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arXiv:2201.07574 · Journal of Computational Electronics

Can Wigner distribution functions with collisions satisfy complete positivity and energy conservation?

Matteo Villani, Xavier Oriols

Published Nov 23, 2021• 5 citations
arXivHTMLPDF

Abstract

Abstract To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modeling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner–Weyl transformation, the modeling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conservation in the electron–photon (electron–phonon) interactions. After comparing collisions models to exact results for an electron interacting with a photon, we conclude that there is no fundamental restriction that prevents a collision model developed within the (reduced) density matrix or Wigner formalisms to satisfy simultaneously complete positivity and energy conservation. However, at the practical level, the development of such satisfactory collision model seems very complicated. Collision models with an explicit knowledge of the microscopic state ascribed to each electron seems recommendable (Bohmian conditional wavefunction), since they allow to model collisions of each electron individually in a controlled way satisfying both complete positivity and energy conservation.

DOI
10.1007/s10825-021-01798-1

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Abstract To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modeling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner–Weyl transformation, the modeling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conservation in the electron–photon (electron–phonon) interactions. After comparing collisions models to exact results for an electron interacting with a photon, we conclude that there is no fundamental restriction that prevents a collision model developed within the (reduced) density matrix or Wigner formalisms to satisfy simultaneously complete positivity and energy conservation. However, at the practical level, the development of such satisfactory collision model seems very complicated. Collision models with an explicit knowledge of the microscopic state ascribed to each electron seems recommendable (Bohmian conditional wavefunction), since they allow to model collisions of each electron individually in a controlled way satisfying both complete positivity and energy conservation.
Abstract We determine the Kirkwood–Dirac quasiprobability (KDQ) distribution associated to the stochastic instances of internal energy variations for the quantum system and environment particles in coherent Markovian collision models. In the case the interactions between the quantum system and the particles do not conserve energy, the KDQ of the non-energy-preserving stochastic work is also derived. These KDQ distributions can account for non-commutativity, and return the unperturbed average values and variances for a generic interaction-time, and generic local initial states of the quantum system and environment particles. Using this nonequilibrium-physics approach, we certify the conditions under which the collision process of the model exhibits quantum traits, and we quantify the rate of energy exchanged by the quantum system by looking at the variance of the KDQ energy distributions. Finally, we propose an experimental test of our results on a superconducting quantum circuit implementing a qubit system, with microwave photons representing the environment particles.
Without access to the full quantum state, modeling dissipation in an open system requires approximations. The physical soundness of such approximations relies on using realistic microscopic models of dissipation that satisfy completely positive dynamical maps. Here we present an approach based on the use of the Bohmian conditional wave function that, by construction, ensures a completely positive dynamical map for either Markovian or non-Markovian scenarios while allowing the implementation of realistic dissipation sources. Our approach is applied to compute the current-voltage characteristic of a resonant tunneling device with a parabolic-band structure, including electron-lattice interactions. A stochastic Schr\"odinger equation is solved for the conditional wave function of each simulated electron. We also extend our approach to (graphenelike) materials with a linear band structure using Bohmian conditional spinors for a stochastic Dirac equation.
Beyond quadratic potentials, classical evolution laws are typically thought to arise from quantum theory as an approximate description. Here we show that classical evolution laws can emerge exactly, provided the environment interacts with the system via collisions. We use this to derive an inequality that lower bounds the distribution of environmental momentum kicks by the absolute value of the Fourier transform of the potential. We show that violating this inequality is necessary and sufficient for a non-quadratic potential to generate negativity from the set of positive Wigner functions. A simple consequence of this is a lower bound on the heating rate for a quantum system to evolve effectively classically. We illustrate our results using the theory of a levitated nanoparticle subject to photon recoil and background gas collisions. We derive the strength of external potential needed to generate Wigner negative states in an optical tweezer and propose an experimental test of whether a nanoparticle in an optical lattice is operating in a non-classical regime.
Electronic decoherence processes in molecules and materials are usually thought and modeled via schemes for the system-bath evolution in which the bath is treated either implicitly or approximately. Here we present computations of the electronic decoherence dynamics of a model many-body molecular system described by the Su-Schrieffer-Heeger Hamiltonian with Hubbard electron-electron interactions using an exact method in which both electronic and nuclear degrees of freedom are taken into account explicitly and fully quantum mechanically. To represent the electron-nuclear Hamiltonian in matrix form and propagate the dynamics, the computations employ the Jordan-Wigner transformation for the fermionic creation/annihilation operators and the discrete variable representation for the nuclear operators. The simulations offer a standard for electronic decoherence that can be used to test approximations. They also provide a useful platform to answer fundamental questions about electronic decoherence that cannot be addressed through approximate or implicit schemes. Specifically, through simulations, we isolate basic mechanisms for electronic coherence loss and demonstrate that electronic decoherence is possible even for one-dimensional nuclear bath. Furthermore, we show that (i) decreasing the mass of the bath generally leads to faster electronic decoherence; (ii) electron-electron interactions strongly affect the electronic decoherence when the electron-nuclear dynamics is not pure-dephasing; (iii) classical bath models with initial conditions sampled from the Wigner distribution accurately capture the short-time electronic decoherence dynamics; (iv) model separable initial superpositions often used to understand decoherence after photoexcitation are only relevant in experiments that employ delta-like laser pulses to initiate the dynamics. These insights can be employed to interpret and properly model coherence phenomena in molecules.
Many-body quantum-mechanical stationary states that have real valued wavefunctions are shown to satisfy a classical conservation of energy equation with a kinetic energy function. The terms in the equation depend on the probability distribution, and, in addition, pressure and velocity functions, but these functions also depend on the probability distribution. There are two possible directions of the velocity that satisfy the energy equation. A linear momentum function is defined that integrates to zero, and this property is consistent with the expectation value of the linear momentum for stationary states with real-valued wave functions. The energy equation is integrated to obtain a version of the well known energy equation involving reduced density matrices, where the kinetic energy functional of the one-particle density matrix is replaced by a function of the electron density and a velocity function. Also, the noninteracting kinetic energy functional from the Hohenberg--Kohn theorem is given as an explicit functional of the orbital densities. For the purpose of describing the behavior of particles in a stationary state, a model based on the energy equation is constructed. The model is evaluated for the two different velocity directions using the grounds state of the particle in a one-dimensional box and the hydrogen atom. For one velocity direction, equations of motions with contradictory properties are obtained, and, in the other, an unstable system is found. A discussion is given with suggestions of additional elements that might improve the model.
In the standard formulation of quantum mechanics states of a system are described by wave functions or density operators . However quantum description of the system can be given in many other ways , for example, in Wigner-Moyal or Feynman formulations . All these representation are equivalent in the the sense of physical results, but the form of presentation of quantum mechanics is different as wave functions, Wigner-Liouville functions or other representations of density operators differ in essential way from each other. In this work we are dealing with Wigner and probability representation of quantum mechanics. The probability representations or tomography representation was recently proposed in terms of marginal distribution functions (MDF). MDF describing the quantum states are positive distribution functions connected with wave functions or density matrices by known integral transformation. Sign conservation of the of the MDF can be valuable in computer simulations to overcome the "sign problem". Important advantage of the probability representation is that the quantum transitions between quantum states are descried by non-negative probabilities. This work is devoted to the development of a new stochastic approaches to numerical solution of the evolution equations for Wigner-Liouviile (WL) and marginal distribution functions. To solve the WL equation we combine both molecular dynamics and Monte Carlo methods and compute traces of the dynamical operators. The results obtained for a system of electrons and random scatterers clearly demonstrate that the many-particle interaction between the electrons leads to an enhancement of the conductivity compared to the case of noninteracting electrons. To obtain evolution of MDF we developed new stochastic approach to solution of the generalized Langevin equation (GLE). GLE is derived from Kolmogorov equations for Green function of evolution equation for MDF. We discuss the basic relations and main ideas of this approach, compare obtained numerical results with results of independent finite-difference calculations for quantum oscillator and quantum particles crossing the finite well and tunneling through the Gaussian barrier.