Wigner Quasiprobability Distribution Function Research Articles

Wigner quasi-probability distribution of a resonator coherent field interacting with a flux qubit via two-photon coupling

The Wigner quasi-probability distribution function is an important tool for estimating the quantumness of quantum states. The Wigner distribution function (WDF) is related monotonically to the coherent field state density operator and their probability distributions. In this paper, we analyze the time-dependent resonator coherent field states produced by a flux qubit’s interaction with a resonator-field through a two-photon coupling. The WDF dynamics of the coherent field states is investigated under the effects of the qubit-resonator detuning and the intrinsic decoherence. The WDF non-classicality associated with the mixedness resonator entropy dynamics can be controlled by varying the qubit-resonator interaction, detuning as well as the decoherence. The intrinsic decoherence clearly affects the WDF non-classicality, identified by the negative values of the WDF. Finally, by fixing one or both of the complex phase space components, we are able to explore the dynamics of the Wigner-distribution functions.

Manipulating light–matter interaction into strong coupling regime for photon entanglement in plasmonic lattices

Enhancing light–matter interaction into the strong coupling regime attracts tremendous attention in both theory and experiment, which presents essential significance in research from nano-optics to quantum information. In this work, the entanglement effect is observed in the photons emitted from a plasmonic lattice as the coherent light–matter interaction occurs into the strong coupling regime with a Rabi splitting of 93.4 meV. A full quantum mechanical treatment based on the number state representation is established to reveal the underlying physics of the strong coupling phenomenon, especially the femtosecond dynamics of energy exchange and damping. The entangled split states display oscillating concurrence and negative Wigner quasiprobability distribution function, which demonstrates that this designed plasmonic lattice system can serve as an on-demand entangled photon source for quantum information.

NMR Hamiltonian as an effective Hamiltonian to generate Schrödinger’s cat states

This report experimentally demonstrates that the theoretical background of the atom–field scenario points out that the NMR quadrupolar Hamiltonian works as an effective Hamiltonian to generate Schrödinger’s cat states in a $$2I+1$$ low-dimensional Hilbert space. The versatility of this nuclear spin setup is verified by monitoring the $$^{23}$$ Na nucleus of a lyotropic liquid crystal sample at the nematic phase. The quantum state tomography and the Wigner quasiprobability distribution function are performed to characterize the accuracy of the experimental implementation.

Classical universe arising from quantum cosmology

In this paper, we study the classical limit and unitary evolution of quantum cosmology by applying the Weyl--Wigner--Groenewold--Moyal formalism of deformation quantization to quantum cosmology of a homogeneous and isotropic universe with positive spatial curvature and conformally coupled scalar field. The corresponding quantum cosmology (similar to the Schr\"{o}dinger interpretation in canonical quantization scheme of quantum cosmology) is described by the Moyal--Wheeler--DeWitt equation which has an exact solution in Moyal phase space, resulting in Wigner quasiprobability distribution function, peaking over the classical solutions. We show that for a large value of the quantum number $n$, the emerged classical universe is filled with radiation with quantum mechanical origin. Also, we introduce a canonical transformation on the scalar field sector of the model such that the conjugate momenta of the new canonical variable appear linearly in the transformed total Hamiltonian. Using this canonical transformation, we show that, it may lead to disentangle the time from the true dynamical variables. We obtain the time-dependent Wigner function for a coherent as well as for squeezed states. We show that the peak of these Wigner functions follows the classical trajectory in the phase space.

Nonlinear coherent states of the para-Bose oscillator and their non-classical features

The construction of nonlinear coherent states via a unitary displacement operator is possible only for a few quantum-mechanical systems. In this paper, we define two non-unitary and a unitary displacement operators with the help of corresponding f -deformed bosonic annihilation and creation operators. While the action of the non-unitary displacement type operators on the vacuum state of field results in two new families of nonlinear coherent states (NLCSs), the unitary displacement operator reproduces Wigner-Heisenberg coherent states of the Gilmore-Perelomov type. We prove that the introduced NLCSs satisfy the resolution of the identities through positive definite measures. We also examine the non-classical properties of the obtained NLCSs by evaluating Klyshko’s criterion, Mandel’s parameter, quadrature squeezing and Wigner quasi-probability distribution function, in detail. Finally, we propose a simple scheme for the physical generation of the introduced NLCSs of the Gilmore-Perelomov type.

Approximate particle number distribution from direct stochastic sampling of the Wigner function

We consider the Wigner quasi-probability distribution function of a single mode of an electromagnetic or matter-wave field to address the question of whether a direct stochastic sampling and binning of the absolute square of the complex field amplitude can yield a distribution function $\tilde{P}_n$ that closely approximates the true particle number probability distribution $P_n$ of the underlying quantum state. By providing an operational definition of the binned distribution $\tilde{P}_n$ in terms of the Wigner function, we explicitly calculate the overlap between $\tilde{P}_n$ and ${P}_n$ and hence quantify the statistical distance between the two distributions. We find that there is indeed a close quantitative correspondence between $\tilde{P}_n$ and $P_n$ for a wide range of quantum states that have smooth and broad Wigner function relative to the scale of oscillations of the Wigner function for the relevant Fock state. However, we also find counterexamples, including states with high mode occupation, for which $\tilde{P}_n$ does not closely approximate $P_n$.

Cavity-Enhanced Second-Order Nonlinear Photonic Logic Circuits

A large obstacle for realizing quantum photonic logic is the weak optical nonlinearity of available materials, which results in large power consumption. In this paper, we present the theoretical design of all-optical logic with second order ($\chi^{(2)}$) nonlinear bimodal cavities and their networks. Using semiclassical models derived from the Wigner quasi-probability distribution function, we analyze the power consumption and signal-to-noise ratio (SNR) of networks implementing an optical AND gate and an optical latch. Comparison between the second and third order $(\chi^{(3)})$ optical logic reveals that while the $\chi^{(3)}$ design outperforms the $\chi^{(2)}$ design in terms of the SNR for the same input power, employing the $\chi^{(3)}$ nonlinearity necessitates the use of cavities with ultra high quality factors ($Q\sim 10^6$) to achieve gate power consumption comparable to that of the $\chi^{(2)}$ design at significantly smaller quality factors ($Q \sim 10^4$). Using realistic estimates of the $\chi^{(2)}$ and $\chi^{(3)}$ nonlinear susceptibilities of available materials, we show that at achievable quality factors ($Q \sim 10^4$), the $\chi^{(2)}$ design is an order of magnitude more energy efficient than the corresponding $\chi^{(3)}$ design.

Generation of bright squeezed light at 1.5 m telecommunication band and its Wigner function reconstruction

The squeezed light at 1.5 m telecommunication band has been considered as an important resource of continuous variable (CV) practical fiber-based quantum information research because it is the lowest loss in fiber. A bright phase quadrature squeezed light for continuous variable at 1.5 m is demonstrated from a semi-monolithic degenerate optical parametric amplifier (DOPA) based on a periodically poled KTiOPO4 (PPKTP) crystal. The laser source is a continuous wave (CW) single-frequency fiber laser at 1.5 m, which is sent through a ring mode cleaner (MC) as a preliminary spatial and noise filter. And then the main portion of the output from the MC is used for external-enhanced second harmonic generation to obtain a CW single-frequency low noise laser at 780 nm that acts as the pump of the DOPA. The residual light of the output from the MC at 1.5 m is used as the injected signal light of the DOPA and the local oscillator (LO) of a balanced homodyne detector (BHD) system. The DOPA is built by using a type-I PPKTP crystal and a piezo-actuated output coupler and works in double-resonance case with a threshold power of 230 mW. When the DOPA is operating in the state of amplification, the output down-conversion field should be a bright phase quadrature squeezed light, where the relative phase between the pump and the injected signal is locked to 0. A 4.7 dB bright phase quadrature squeezed light is measured by the BHD system with the pump light of 110 mW and the injected signal of 3 mW, where the relative phase between the down-conversion field and the LO is locked to 0. Our measurement is limited by the optical losses and the detection efficiency. We have taken into account the detection efficiency of 86.6%, and the actual squeezing of the squeezed light being 6.3 dB. Moreover, because it is so crucial a process for CV quantum information system that the transmission and evolution of the CV squeezed states in the fiber may reappear in all information of the quantum states in the phase space, then the bright squeezed light is detected by a BHD system in the time domain, and its Wigner quasi-probability distribution function can be reconstructed by using a quantum tomographic technique. Furthermore, the bright squeezed state at 1.5 m is an ideal source for fiber-based long-distance quantum information because of its stability and bright mean field.

Dynamics of physical properties of a single-mode quantized field non-linearly and non-resonantly interacting with two V-type three-level atoms passing consecutively through a cavity

In this paper we address the analytical solution of the non-resonant interaction between two identical V-type three-level atoms passing consecutively through a single-mode cavity field in the presence of intensity-dependent coupling. By considering an identical initial condition for both atoms and an initial coherent field, we find the analytical solution of the state vector of the entire atom–field system. Accordingly, we could carefully investigate the influence of various parameters in the circumstances of the interacting system on different physical quantities such as the atomic population inversion, atom–field entanglement, field squeezing, sub-Poissonian statistics and the Wigner quasi-probability distribution function. In detail, we discuss numerically the influences of the detuning parameters and a particular nonlinearity function on the mentioned quantities and demonstrate that they have substantial effects on the temporal behavior of the above-mentioned non-classical properties.

Algebraic and group treatments to nonlinear displaced number states and their nonclassicality features: A new approach

Recently, nonlinear displaced number states (NDNSs) have been manually introduced, in which the deformation function f(n) has been artificially added to the previously well-known displaced number states (DNSs). Indeed, just a simple comparison has been performed between the standard coherent state and nonlinear coherent state for the formation of NDNSs. In the present paper, after expressing enough physical motivation of our procedure, four distinct classes of NDNSs are presented by applying algebraic and group treatments. To achieve this purpose, by considering the DNSs and recalling the nonlinear coherent states formalism, the NDNSs are logically defined through an algebraic consideration. In addition, by using a particular class of Gilmore–Perelomov-type of SU(1, 1) and a class of SU(2) coherent states, the NDNSs are introduced via group-theoretical approach. Then, in order to examine the nonclassical behavior of these states, sub-Poissonian statistics by evaluating Mandel parameter and Wigner quasi-probability distribution function associated with the obtained NDNSs are discussed, in detail.

Entanglement with negative Wigner function of almost 3,000 atoms heralded by one photon.

Quantum-mechanically correlated (entangled) states of many particles are of interest in quantum information, quantum computing and quantum metrology. Metrologically useful entangled states of large atomic ensembles have been experimentally realized, but these states display Gaussian spin distribution functions with a non-negative Wigner quasiprobability distribution function. Non-Gaussian entangled states have been produced in small ensembles of ions, and very recently in large atomic ensembles. Here we generate entanglement in a large atomic ensemble via an interaction with a very weak laser pulse; remarkably, the detection of a single photon prepares several thousand atoms in an entangled state. We reconstruct a negative-valued Wigner function--an important hallmark of non-classicality--and verify an entanglement depth (the minimum number of mutually entangled atoms) of 2,910 ± 190 out of 3,100 atoms. Attaining such a negative Wigner function and the mutual entanglement of virtually all atoms is unprecedented for an ensemble containing more than a few particles. Although the achieved purity of the state is slightly below the threshold for entanglement-induced metrological gain, further technical improvement should allow the generation of states that surpass this threshold, and of more complex Schrödinger cat states for quantum metrology and information processing. More generally, our results demonstrate the power of heralded methods for entanglement generation, and illustrate how the information contained in a single photon can drastically alter the quantum state of a large system.

Spin squeezing in a quadrupolar nuclei NMR system.

We have produced and characterized spin-squeezed states at a temperature of 26 °C in a nuclear magnetic resonance quadrupolar system. The experiment was carried out on 133Cs nuclei of spin I=7/2 in a sample of lyotropic liquid crystal. The source of spin squeezing was identified as the interaction between the quadrupole moment of the nuclei and the electric field gradients present within the molecules. We use the spin angular momentum representation to describe formally the nonlinear operators that produce the spin squeezing on a Hilbert space of dimension 2I+1=8. The quantitative and qualitative characterization of this spin-squeezing phenomenon is expressed by a squeezing parameter and squeezing angle developed for the two-mode Bose-Einstein condensate system, as well as by the Wigner quasiprobability distribution function. The generality of the present experimental scheme points to potential applications in solid-state physics.

高阶TEM01模压缩态Wigner准概率分布函数的量子重构

高阶TEM₀₁模压缩态Wigner准概率分布函数的量子重构

Entanglement of a quantum optical elliptic vortex

We calculate the entanglement of a generalized elliptical vortex formed by quantized radiation field, using Wigner quasiprobability distribution function for such states. We find a critical squeezing parameter above which the entanglement is less for higher vorticity, which is counter intuitive.

Wigner distribution of elliptical quantum optical vortex

We calculate the Wigner quasiprobability distribution function of the quantum elliptical vortex in elliptical beam (EEV), produced by coupling squeezed coherent states of two modes. The coupling between the two modes is performed by using beam splitter (BS) or a dual channel directional coupler (DCDC). The quantum interference due to the coupling between the two modes promises the generation of controlled entanglement for quantum computation and quantum tomography.

Generation of squeezed vacuum states by PPKTP crystal and its Wigner quasi-probability distribution function reconstruction

We demonstrated experimentally the generation of squeezed vacuum by a periodically poled KTP crystal in a continuous-wave optical parametric oscillator，which was pumped with 532nm field. The squeezed vacuum of 3.41 dB at 1064 nm was detected by a homodyne detection system. Moreover，the Wigner quasi-probability distribution function of the squeezed vacuum was reconstructed using quantum tomography technique.

A new form of Wigner functions on the noncommutative space

Wigner quasi-probability distribution function in phase space is a special (Weyl) representation of the density matrix. It has been very useful in many areas of quantum mechanics. Starting from fundamental principle of the Weyl correspondence, we derive explicit form of the Wigner function (WF) for noncommutative quantum mechanics (NCQM), and prove that it satisfies a generalized *-genvalue equation. We also give some examples to show that the new form of WF indeed has this property.

Wigner functions for curved spaces. I. On hyperboloids

We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature, in this article on hyperboloids, which returns the correct marginals and has the covariance of the Shapiro functions under SO(D,1) transformations. To the free systems obeying the Laplace–Beltrami equation on the hyperboloid, we add a conic-oscillator potential in the hyperbolic coordinate. As an example, we analyze the one-dimensional case on a hyperbola branch, where this conic-oscillator is the Pöschl–Teller potential. We present the analytical solutions and plot the computed results. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.

DEFORMATION QUANTIZATION: QUANTUM MECHANICS LIVES AND WORKS IN PHASE-SPACE

Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (e.g. quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides — coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.

Connection between two Wigner functions for spin systems

In 1981, Agarwal proposed a Wigner quasiprobability distribution function on the group SU(2) that serves to analyze two-particle spin states on a sphere. Recent work by our group has included the definition of an apparently distinct Wigner function on generic Lie groups whose natural range has the dimension of the group and serves for all square-integrable representations; for the SO(3) case this entails a three-dimensional ``meta-phase'' space. Both have the fundamental properties covariance and completeness. Here we show how the former is obtained as a restriction of the latter.