Wigner Phase-space Distribution Research Articles

Quantum Sensitivity of a Photon-added Molecular Wave Packet

Abstract We study the temporal dynamics of a photon-added SU(2) coherent state, involving vibrational bound states of an Iodine molecule. The nonclassicality of Schrodinger cat-like and compass-like photon-added states of this anharmonic potential is explored through Mandel Q parameter and the negative region in their phase spaces. The Wigner phase space distribution is thoroughly studied for the photon added molecular state and displayed for cat- and compass-like states. Generation and control of the sub-Planck interference structures become intriguing with photon addition. We ensure that the photon addition helps to improve the quantum sensitivity for a diatomic molecular system by comparing it between photon-added and photon-subtracted states. The detailed study of quantum sensitivity reveals that, one can use a photon-added molecular wave packet as a probe to attain sensitivity to displacement, that greatly exceeds the standard quantum limit. We also report that the nature of variation of the quantum sensitivity becomes qualitatively commensurate with that of the Wigner-negativity.

New angular momentum coherent states via the nonlinear Bose SU(2) generators and their nonclassical behaviors

I theoretically construct a kind of new angular momentum coherent states by virtue of the nonlinear Bose SU(2) generators, and investigate their nonclassicality by analyzing the sub-Poissonian distributions, photon number distributions, entanglement entropy and Wigner phase-space distributions. The results show that the new states always have the sub-Poissonian distributions for all of the photon number sum [Formula: see text] of two modes, the photon number distributions are subjected to the parameter q, and the entanglement always increases quickly and then decreases slowly with increasing [Formula: see text], and the new states for odd q possess more stronger nonclassicality than for even q.

Phase-space quantum distributions and information theory

The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.

Direct Characteristic-Function Tomography of Quantum States of the Trapped-Ion Motional Oscillator.

We implement direct readout of the symmetric characteristic function of quantum states of the motional oscillation of a trapped calcium ion. Suitably chosen internal state rotations combined with internal state-dependent displacements, based on bichromatic laser fields, map the expectation value of the real or imaginary part of the displacement operator to the internal states, which are subsequently read out. Combining these results provides full information about the symmetric characteristic function. We characterize the technique by applying it to a range of archetypal quantum oscillator states, including displaced and squeezed Gaussian states as well as two and three component superpositions of displaced squeezed states. For each, we discuss relevant features of the characteristic function and Wigner phase-space quasiprobability distribution. The direct reconstruction of these highly nonclassical oscillator states using a reduced number of measurements is an essential tool for understanding and optimizing the control of oscillator systems for quantum sensing and quantum information applications.

Finite-dimensional pair coherent state engendered via the nonlinear Bose operator realization and its Wigner phase-space distributions

We theoretically analyze the photon number distribution, entanglement entropy, and Wigner phase-space distribution, considering the finite-dimensional pair coherent state (FDPCS) generated in the nonlinear Bose operator realization. Our results show that the photon number distribution is governed by the two-mode photon number sum q of the FDPCS, the entanglement of the FDPCS always increases quickly at first and then decreases slowly for any q, and the nonclassicality of the FDPCS for odd q is more stronger than that for even q.

Phase Space Analysis of the Two-mode Binomial State Produced by Quantum Entanglement in a Beamsplitter

Phase space analysis of quantum states is a newly developed topic in quantum optics. In this work we present Wigner phase space distributions for the two-mode binomial state produced by quantum entanglement between a vacuum state and a number state in a beamsplitter. By using two new binomial formulas involving two-variable Hermite polynomials and the so-called entangled Wigner operator, we find that the analytical Wigner function for the binomial state |ξ〉q ≡ D(ξ) |q, 0〉 is related to a Laguerre polynomial, i.e., $ W\left (\sigma _{,}\gamma \right ) =\frac {(-1)^{q}e^{-\left \vert \gamma \right \vert ^{2}-\left \vert \sigma \right \vert ^{2}}}{\pi ^{2}}L_{q}\left (\left \vert \frac {-\varsigma (\sigma -\gamma )+\sigma ^{\ast }+\gamma ^{\ast }} {\sqrt {1+|\varsigma |^{2}}}\right \vert ^{2}\right ) $ and its marginal distributions are proportional to the module-square of a single-variable Hermite polynomial. Also, the numerical results show that the larger number sum q of two modes lead to the stronger interference effect and the nonclassicality of the states |ξ〉q is stronger for odd q than for even q.

Constructing phase space distributions with internal symmetries

We discuss an ab initio world-line approach to constructing phase space distributions in systems with internal symmetries. Starting from the Schwinger-Keldysh real time path integral in quantum field theory, we derive the most general extension of the Wigner phase space distribution to include color and spin degrees of freedom in terms of dynamical Grassmann variables. The corresponding Liouville distribution for colored particles, which obey Wong's equation, has only singlet and octet components, while higher moments are fully constrained by the Grassmann algebra. The extension of phase space dynamics to spin is represented by a generalization of the Pauli-Lubanski vector; its time evolution via the Bargmann-Michel-Telegdi equation also follows from the phase space trajectories of the underlying Grassmann coordinates. Our results for the Liouville phase space distribution in systems with both spin and color are of interest in fields as diverse as chiral fluids, finite temperature field theory and polarized parton distribution functions. We also comment on the role of the chiral anomaly in the phase space dynamics of spinning particles.

Formulation of state projected centroid molecular dynamics: Microcanonical ensemble and connection to the Wigner distribution.

A derivation of quantum statistical mechanics based on the concept of a Feynman path centroid is presented for the case of generalized density operators using the projected density operator formalism of Blinov and Roy [J. Chem. Phys. 115, 7822-7831 (2001)]. The resulting centroid densities, centroid symbols, and centroid correlation functions are formulated and analyzed in the context of the canonical equilibrium picture of Jang and Voth [J. Chem. Phys. 111, 2357-2370 (1999)]. The case where the density operator projects onto a particular energy eigenstate of the system is discussed, and it is shown that one can extract microcanonical dynamical information from double Kubo transformed correlation functions. It is also shown that the proposed projection operator approach can be used to formally connect the centroid and Wigner phase-space distributions in the zero reciprocal temperature β limit. A Centroid Molecular Dynamics (CMD) approximation to the state-projected exact quantum dynamics is proposed and proven to be exact in the harmonic limit. The state projected CMD method is also tested numerically for a quartic oscillator and a double-well potential and found to be more accurate than canonical CMD. In the case of a ground state projection, this method can resolve tunnelling splittings of the double well problem in the higher barrier regime where canonical CMD fails. Finally, the state-projected CMD framework is cast in a path integral form.

Multifunctional interleaved geometric-phase dielectric metasurfaces

Shared-aperture technology for multifunctional planar systems, performing several simultaneous tasks, was first introduced in the field of radar antennas. In photonics, effective control of the electromagnetic response can be achieved by a geometric-phase mechanism implemented within a metasurface, enabling spin-controlled phase modulation. The synthesis of the shared-aperture and geometric-phase concepts facilitates the generation of multifunctional metasurfaces. Here shared-aperture geometric-phase metasurfaces were realized via the interleaving of sparse antenna sub-arrays, forming Si-based devices consisting of multiplexed geometric-phase profiles. We study the performance limitations of interleaved nanoantenna arrays by means of a Wigner phase-space distribution to establish the ultimate information capacity of a metasurface-based photonic system. Within these limitations, we present multifunctional spin-dependent dielectric metasurfaces, and demonstrate multiple-beam technology for optical rotation sensing. We also demonstrate the possibility of achieving complete real-time control and measurement of the fundamental, intrinsic properties of light, including frequency, polarization and orbital angular momentum.

Parton transverse momentum and orbital angular momentum distributions

The quark orbital angular momentum component of proton spin, $L_q$, can be defined in QCD as the integral of a Wigner phase space distribution weighting the cross product of the quark's transverse position and momentum. It can also be independently defined from the operator product expansion for the off-forward Compton amplitude in terms of a twist-three generalized parton distribution. We provide an explicit link between the two definitions, connecting them through their dependence on partonic intrinsic transverse momentum. Connecting the definitions provides the key for correlating direct experimental determinations of $L_q$, and evaluations through Lattice QCD calculations. The direct observation of quark orbital angular momentum does not require transverse spin polarization, but can occur using longitudinally polarized targets.

Observation of Quantum Interference between Separated Mechanical Oscillator Wave Packets.

We directly observe the quantum interference between two well-separated trapped-ion mechanical oscillator wave packets. The superposed state is created from a spin-motion entangled state using a heralded measurement. Wave packet interference is observed through the energy eigenstate populations. We reconstruct the Wigner function of these states by introducing probe Hamiltonians which measure Fock state populations in displaced and squeezed bases. Squeezed-basis measurements with 8dB squeezing allow the measurement of interference for Δα=15.6, corresponding to a distance of 240nm between the two superposed wave packets.

Wigner phase space distribution via classical adiabatic switching.

Evaluation of the Wigner phase space density for systems of many degrees of freedom presents an extremely demanding task because of the oscillatory nature of the Fourier-type integral. We propose a simple and efficient, approximate procedure for generating the Wigner distribution that avoids the computational difficulties associated with the Wigner transform. Starting from a suitable zeroth-order Hamiltonian, for which the Wigner density is available (either analytically or numerically), the phase space distribution is propagated in time via classical trajectories, while the perturbation is gradually switched on. According to the classical adiabatic theorem, each trajectory maintains a constant action if the perturbation is switched on infinitely slowly. We show that the adiabatic switching procedure produces the exact Wigner density for harmonic oscillator eigenstates and also for eigenstates of anharmonic Hamiltonians within the Wentzel-Kramers-Brillouin (WKB) approximation. We generalize the approach to finite temperature by introducing a density rescaling factor that depends on the energy of each trajectory. Time-dependent properties are obtained simply by continuing the integration of each trajectory under the full target Hamiltonian. Further, by construction, the generated approximate Wigner distribution is invariant under classical propagation, and thus, thermodynamic properties are strictly preserved. Numerical tests on one-dimensional and dissipative systems indicate that the method produces results in very good agreement with those obtained by full quantum mechanical methods over a wide temperature range. The method is simple and efficient, as it requires no input besides the force fields required for classical trajectory integration, and is ideal for use in quasiclassical trajectory calculations.

Wigner phase-space distribution as a wave function

We demonstrate that the Wigner function of a pure quantum state is a wave function in a specially tuned Dirac bra-ket formalism and argue that the Wigner function is in fact a probability amplitude for the quantum particle to be at a certain point of the classical phase space. Additionally, we establish that in the classical limit, the Wigner function transforms into a classical Koopman-von Neumann wave function rather than into a classical probability distribution. Since probability amplitude need not be positive, our findings provide an alternative outlook on the Wigner function's negativity.

Five momenta

A quantum or classical wavefunction depending on position can be associated with a local momentum in at least five apparently different ways: first, as the phase gradient of the wavefunction; second, as the local expectation value of the momentum operator; third, via the local current; fourth, via the Wigner phase-space distribution function; and fifth, as the weak value of momentum with position postselected. The different formulas are all equivalent, but give different insights into the underlying physics. Momenta one through three are largely familiar, but the fourth and fifth are less so.

Wigner Function for Klein—Gordon Landau Problem

First we calculate the Wigner phase-space distribution function for the Klein—Gordan Landau problem on a commmutative space. Then we study the modifications introduced by the coordinate-coordinate noncommuting and momentum-momentum noncommuting, namely, by using a generalized Bopp's shift method we construct the Wigner function for the Klein—Gordan Landau problem both on a noncommutative space (NCS) and a noncommutative phase space (NCPS).

Optimally focusing wave packets

An appropriately prepared real-valued wave packet moving in one space dimension will focus during a brief period of time even in the absence of any force. We illustrate this phenomenon by considering the time evolution of the elementary superposition of the ground state and the second excited state of a harmonic oscillator. Moreover, we show that a variation of the superposition parameter leads us from a domain of enhanced spreading via a point of suppressed spreading to a region where the wave packets focuses before it spreads again. We determine the points of maximal spreading and optimal focusing. Our analysis of this unusual behavior of a free quantum particle rests on the time dependence of (i) the average separation of the wave packet from the origin, (ii) the probability density in position space, and (iii) the Wigner phase space distribution. We conclude our search for optimally focusing wave packets by solving the corresponding variational problem with respect to a family of measures expressing the width of the wave packet.

Quantum Smoluchowski equation for driven systems

We consider a driven quantum harmonic oscillator strongly coupled to a heat bath. Starting from the exact quantum Langevin equation, we use a Green's function approach to determine the corresponding semiclassical equation for the Wigner phase space distribution. In the limit of high friction and high temperature, we apply Brinkman's method to derive the quantum Smoluchowski equation for the probability distribution in position space. We further determine the range of validity of the equation and discuss the special case of a Brownian parametric oscillator.

Closure of quantum hydrodynamic moment equations

The hydrodynamic formulation of mixed quantum states involves a hierarchy of coupled equations of motion for the momentum moments of the Wigner function. In this work a closure scheme for the hierarchy is developed. The closure scheme uses information contained in the lower known moments to expand the Wigner phase-space distribution function in a Gauss-Hermite orthonormal basis. The higher moment required to terminate the hierarchy is then easily obtained from the reconstructed approximate Wigner function by a straightforward integration over the momentum space. Application of the moment closure scheme is demonstrated for the dissipative and nondissipative dynamics of two different systems: (i) double-well potential, (ii) periodic potential.

Phase space description of spin dynamics

The spin system with Hamiltonian (ξ and σ are external and internal field parameters) is treated as an example of the phase space description of spin dynamics using a master equation for the quasiprobability distribution function of spin orientations in the representation (phase) space of the polar angles (analogous to the Wigner phase space distribution for translational motion). The master equation yields (via the Wigner–Stratonovich transformation of the density matrix) the solution as a Fourier series in the spherical harmonics with Fourier coefficients given by the statistical moments in a manner analogous to the classical distribution. The relaxation time of the longitudinal component of the spin can be estimated using a quantum generalization of the classical integral relaxation time via the stationary distribution and diffusion coefficient of the master equation.

Phase-space formulation of the nonlinear longitudinal relaxation of the magnetization in quantum spin systems

Nonlinear longitudinal relaxation of a spin in a uniform external dc magnetic field is treated using a master equation for the quasiprobability distribution function of spin orientations in the configuration space of polar and azimuthal angles (analogous to the Wigner phase space distribution for translational motion). The solution of the corresponding classical problem of the rotational Brownian motion of a magnetic moment in an external magnetic field essentially carries over to the quantum regime yielding in closed form the dependence of the longitudinal spin relaxation on the spin size S as well as an expression for the integral relaxation time, which in linear response reduces to that previously given by D. A. Garanin [Phys. Rev. E 55, 2569 (1997)] using the density matrix approach. The nonlinear relaxation is dominated by a single exponential having as time constant the integral relaxation time. Thus a simple description in terms of a Bloch equation holds even for the nonlinear response of a giant spin.