Von Neumann Equation Research Articles

Performance evaluation of numerical methods for the Maxwell–Liouville–von Neumann equations

The Maxwell–Liouville–von Neumann (MLN) equations are a valuable tool in nonlinear optics in general and to model quantum cascade lasers in particular. Several numerical methods to solve these equations with different accuracy and computational complexity have been proposed in related literature. We present an open-source framework for solving the MLN equations and parallel implementations of three numerical methods using OpenMP. The performance measurements demonstrate the efficiency of the parallelization.

On the reduced dynamics of a subset of interacting bosonic particles

The quantum dynamics of a subset of interacting bosons in a subspace of fixed particle number is described in terms of symmetrized many-particle states. A suitable partial trace operation over the von Neumann equation of an N-particle system produces a hierarchical expansion for the subdynamics of M≤N particles. Truncating this hierarchy with a pure product state ansatz yields the general, nonlinear coherent mean-field equation of motion. In the special case of a contact interaction potential, this reproduces the Gross–Pitaevskii equation. To account for incoherent effects on top of the mean-field evolution, we discuss possible extensions towards a second-order perturbation theory that accounts for interaction-induced decoherence in form of a nonlinear Lindblad-type master equation.

Quantum mechanics on phase space: The hydrogen atom and its Wigner functions

Symplectic quantum mechanics (SQM) considers a non-commutative algebra of functions on a phase space Γ and an associated Hilbert space HΓ, to construct a unitary representation for the Galilei group. From this unitary representation the Schrödinger equation is rewritten in phase space variables and the Wigner function can be derived without the use of the Liouville–von Neumann equation. In this article the Coulomb potential in three dimensions (3D) is resolved completely by using the phase space Schrödinger equation. The Kustaanheimo–Stiefel(KS) transformation is applied and the Coulomb and harmonic oscillator potentials are connected. In this context we determine the energy levels, the amplitude of probability in phase space and correspondent Wigner quasi-distribution functions of the 3D-hydrogen atom described by Schrödinger equation in phase space.

Electrical Conductivity of Charged Particle Systems and Zubarev’s Nonequilibrium Statistical Operator Method

One of the fundamental problems in physics which are not rigorously solved yet is the statistical mechanics of nonequilibrium processes. An important contribution to describe irreversible behavior starting from reversible Hamiltonian dynamics was given by D. N. Zubarev who invented the method of the nonequilibrium statistical operator (NSO). We discuss this approach, in particular the extended von Neumann equation, and consider as example the electrical conductivity of a charged particle system. The selection of the set of relevant observables is considered. The relation between kinetic theory and linear response theory is shown. Using thermodynamic Green functions, a systematic treatment of correlation functions is given, but convergence has to be investigated. Different expressions for the conductivity are compared, and open questions are identified.

Supersymmetric symplectic quantum mechanics

Symplectic Quantum Mechanics SQM considers a non-commutative algebra of functions on a phase space Γ and an associated Hilbert space HΓ to construct a unitary representation for the Galilei group. From this unitary representation the Schrödinger equation is rewritten in phase space variables and the Wigner function can be derived without the use of the Liouville–von Neumann equation. In this article we extend the methods of supersymmetric quantum mechanics SUSYQM to SQM. With the purpose of applications in quantum systems, the factorization method of the quantum mechanical formalism is then set within supersymmetric SQM. A hierarchy of simpler hamiltonians is generated leading to new computation tools for solving the eigenvalue problem in SQM. We illustrate the results by computing the states and spectra of the problem of a charged particle in a homogeneous magnetic field as well as the corresponding Wigner function.

Estimations of the errors between the evolving states generated by two Hamiltonians with the same initial state

Time evolution of a quantum system is described by Schrodinger equation with initial pure state, or von Neumann equation with initial mixed state. In this paper, we estimate the error between the evolving states generated by two Hamiltonians with the same initial pure state. Secondly, according to the method of operator–vector correspondence, we give a relation of the Schrodinger equation and von Neumann equation and then estimate the error between the evolving states generated by two Hamiltonians with the same initial mixed state.

Single-cone finite difference scheme for the (2+1)D Dirac von Neumann equation

An explicit finite difference scheme is presented for the von Neumann equation for (2+1)D Dirac fermions. It is founded upon a staggered space–time grid which ensures a single-cone energy dispersion and performs the time-derivative in one sweep using a three-step leap-frog procedure. It enables a space–time-resolved numerical treatment of the mixed-state dynamics of Dirac fermions within the effective single-particle density matrix formalism. Energy-momentum dispersion, stability and convergence properties are derived. Elementary numerical tests to demonstrate stability properties use parameters which pertain to topological insulator surface states. A method for the simulation of charge injection from an electric contact is presented and tested numerically. Potential extensions of the scheme to a Dirac–Lindblad equation, real-space–time Green's function formulations, and higher-order finite-difference schemes are discussed.

On corrected formula for irradiated graphene quantum conductivity

Graphene membrane irradiated by weak activating periodic electric field in terahertz range is considered. The corrected formula for the graphene quantum conductivity is found. The obtained formula gives complex conjugate results when radiation polarization direction is clockwise or it is opposite clockwise. The found formula allows us to see that the graphene membrane is an oscillating contour. Its eigen frequency coincides with a singularity point of the conductivity and depends on the electrons concentration. So the graphene membrane could be used as an antenna or a transistor and its eigen frequency could be tuned by doping in a large terahertz-infrared frequency range. The obtained formula allows us also to calculate the graphene membrane quantum inductivity and capacitance. The found dependence on electrons concentration is consistent with experiments. The method of the proof is based on study of the time-dependent density matrix. The exact solution of von Neumann equation for density matrix is found for our case in linear approximation on the external field. On this basis the induced current is studied and then the formula for quantum conductivity as a function of external field frequency and temperature is obtained. The method of the proof suggested in this paper could be used to study other problems. The found formula for quantum conductivity can be used to correct the SPPs Dispersion Relation and for the description of radiation process. It would be useful to take the obtained results into account when constructing devices containing graphene membrane nanoantenna. Such project could make it possible to create wireless communications among nanosystems. This would be promising research area of energy harvesting applications.

Extremal Density Matrices for the Expectation Value of a Qudit Hamiltonian

An algebraic procedure to find extremal density matrices for the expectation value of a finite Hamiltonian matrix is established. The extremal density matrices for pure states provide a complete description of the system, that is, its corresponding energy spectrum and projectors. For density matrices representing mixed states, one gets the most probable eigenstates that yield extremal mean values of the energy. The procedure uses mainly the stationary solutions of the von Neumann equation of motion, the orbits of the Hamiltonian, and the positivity conditions of the density matrix. The method is illustrated for matrix Hamiltonians of dimensions d = 2 and d = 3.

Currents and fluctuations of quantum heat transport in harmonic chains

Heat transport in open quantum systems is particularly susceptible to the modeling of system–reservoir interactions. It thus requires us to consistently treat the coupling between a quantum system and its environment. While perturbative approaches are successfully used in fields like quantum optics and quantum information, they reveal deficiencies—typically in the context of thermodynamics, when it is essential to respect additional criteria such as fluctuation-dissipation theorems. We use a non-perturbative approach for quantum dissipative dynamics based on a stochastic Liouville–von Neumann equation to provide a very general and extremely efficient formalism for heat currents and their correlations in open harmonic chains. Specific results are derived not only for first- but also for second-order moments, which requires us to account for both real and imaginary parts of bath–bath correlation functions. Spatiotemporal patterns are compared with weak coupling calculations. The regime of stronger system–reservoir couplings gives rise to an intimate interplay between reservoir fluctuations and heat transfer far from equilibrium.

Quantum mechanics on phase space and the Coulomb potential

Symplectic quantum mechanics (SMQ) makes possible to derive the Wigner function without the use of the Liouville–von Neumann equation. In this formulation of the quantum theory the Galilei Lie algebra is constructed using the Weyl (or star) product with Qˆ=q⋆=q+iħ2∂p,Pˆ=p⋆=p−iħ2∂q, and the Schrödinger equation is rewritten in phase space; in consequence physical applications involving the Coulomb potential present some specific difficulties. Within this context, in order to treat the Schrödinger equation in phase space, a procedure based on the Levi-Civita (or Bohlin) transformation is presented and applied to two-dimensional (2D) hydrogen atom. Amplitudes of probability in phase space and the correspondent Wigner quasi-distribution functions are derived and discussed.

Liquid-state paramagnetic relaxation from first principles

We simulate nuclear and electron spin relaxation rates in a paramagnetic system from first principles. Sampling a molecular dynamics trajectory with quantum-chemical calculations produces a time series of the instantaneous parameters of the relevant spin Hamiltonian. The Hamiltonians are, in turn, used to numerically solve the Liouville--von Neumann equation for the time evolution of the spin density matrix. We demonstrate the approach by studying the aqueous solution of the ${\mathrm{Ni}}^{2+}$ ion. Taking advantage of Kubo's theory, the spin-lattice $({T}_{1})$ and spin-spin $({T}_{2})$ relaxation rates are extracted from the simulations of the time dependence of the longitudinal and transverse magnetization, respectively. Good agreement with the available experimental data is obtained by the method.

Derivation of Markovian Master Equation by Renormalization Group Method

We present a derivation of the Markovian master equation by the renormalization group method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then by applying the renormalization group method, we show that the dependence of the specific initial time on the perturbative solution can be removed and the Markovian semigroup master equation in the Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) form is obtained in the weak coupling limit.

Derivation of a time dependent Schrödinger equation as the quantum mechanical Landau–Lifshitz–Bloch equation

The derivation of the time dependent Schrödinger equation with transversal and longitudinal relaxation, as the quantum mechanical analog of the classical Landau–Lifshitz–Bloch equation, has been described. Starting from the classical Landau–Lifshitz–Bloch equation the transition to quantum mechanics has been performed and the corresponding von-Neumann equation deduced. In a second step the time Schrödinger equation has been derived. Analytical proofs and computer simulations show the correctness and applicability of the derived Schrödinger equation.

Fokker-Planck formalism in magnetic resonance simulations

This paper presents an overview of the Fokker-Planck formalism for non-biological magnetic resonance simulations, describes its existing applications and proposes some novel ones. The most attractive feature of Fokker-Planck theory compared to the commonly used Liouville - von Neumann equation is that, for all relevant types of spatial dynamics (spinning, diffusion, stationary flow, etc.), the corresponding Fokker-Planck Hamiltonian is time-independent. Many difficult NMR, EPR and MRI simulation problems (multiple rotation NMR, ultrafast NMR, gradient-based zero-quantum filters, diffusion and flow NMR, off-resonance soft microwave pulses in EPR, spin-spin coupling effects in MRI, etc.) are simplified significantly in Fokker-Planck space. The paper also summarises the author’s experiences with writing and using the corresponding modules of the Spinach library – the methods described below have enabled a large variety of simulations previously considered too complicated for routine practical use.

Self-consistent scheme for optical response of large hybrid networks of semiconductor quantum dots and plasmonic metal nanoparticles

We discuss a self-consistent scheme for treating the optical response of large, hybrid networks of semiconducting quantum dots (SQDs) and plasmonic metallic nanoparticles (MNPs). Our method is efficient and scalable and becomes exact in the limiting case of weakly interacting SQDs. The self-consistent equations obtained for the steady state are analogous to the von Neumann equations of motion for the density matrix of a SQD placed in an effective electric field computed within the discrete dipole approximation. Illustrative applications of the theory to square and honeycomb SQD, MNP, and hybrid SDQ-MNP lattices as well as SQD-MNP dimers are presented. Our results demonstrate that hybrid SQD-MNP lattices can provide flexible platforms for light manipulation with tunable resonant characteristics.

Theoretical study on ultrafast dynamics of coherent acoustic phonons in semiconductor nanocrystals

We present a theoretical study on the ultrafast dynamics of coherent acoustic phonons in semiconductor quantum dots using continuum model calculations. The excitonic states and the coherent acoustic vibrational modes of semiconductor quantum dots are calculated using the effective mass approximation and continuum elastic medium model, respectively. By solving the Liouville–von Neumann equation and the equation of motion, we obtain the oscillation of coherent acoustic phonon amplitude excited by a pump pulse laser. Owing to the ultrafast excitation of coherent phonons, both the amplitude and the phase of the coherent phonon oscillation are constant with time. This coherent phonon oscillation results in conservation of the coherence of the exciton state, which cannot exist in a system interacting with incoherent phonons. We further study the amplitude and the period of coherent acoustic phonon oscillation as a function of pump pulse energy detuning, quantum dot size, and material.

Real-space finite difference scheme for the von Neumann equation with the Dirac Hamiltonian

A finite difference scheme for the numerical treatment of the von Neumann equation for the (2+1)D Dirac Hamiltonian is presented. It is based on a sequential left–right (ket–bra) application of a staggered space–time scheme for the pure-state Dirac equation and offers a numerical treatment of the general mixed-state dynamics of an isolated quantum system within the von Neumann equation. Thereby this direct scheme inherits all the favorable features of the finite-difference scheme for the pure-state Dirac equation, such as the single-cone energy–momentum dispersion, convergence conditions, and scaling behavior. A conserved functional is identified. Moreover this scheme is shown to conserve both Hermiticity and positivity. Numerical tests comprise a numerical analysis of stability, as well as the simulation of a mixed-state time-evolution of Gaussian wave functions, illustrating Zitterbewegung and transverse current oscillations. Imaginary-potential absorbing boundary conditions and parameters which pertain to topological insulator surface states were used in the numerical simulations.

Arbitrary phase shift of a semiconductor quantum dot charge qubit on a short time scale

We present an experimental technique to better control the phase of an electron charge qubit, formed by a GaAs double quantum dot. A standard non-adiabatic gate pulse is used to generate the qubit rotation around the x-axis of the Bloch sphere. To gain good control of the z-rotation (the phase) on a short time scale, a fast 130 ps tipping pulse is superimposed on the non-adiabatic pulse. The two-axis gate operation is exhibited in the composite pulse excited qubit evolution spectrum. We demonstrate that the dynamic phase can be varied continuously from 0 to by varying the amplitude of the tipping pulse. The understanding of the spectrum is validated through simulation of the von Neumann equation.

Entropy Associated with Information Storage and Its Retrieval

We provide an entropy analysis for light storage and light retrieval. In this analysis, entropy extraction and reduction in a typical light storage experiment are identified. The spatiotemporal behavior of entropy is presented for D1 transition in cold sodium atoms. The governing equations are the reduced Maxwell field equations and the Liouville–von Neumann equation for the density matrix of the dressed atom.