Heisenberg Equations Of Motion Research Articles

Equivalence principle for quantum mechanics in the Heisenberg picture

We present an exact quantum observable analog of the weak equivalence principle for a ‘relativistic’ quantum particle. The quantum geodesic equations are obtained from Heisenberg equations of motion as an exact analog of a fully covariant classical Hamiltonian evolution picture, with the proper identification of the canonical momentum variables as p µ , rather than p µ . We discuss the meaning of the equations in relation to projective measurements as well as equations with solution curves as ones in the noncommutative geometric picture of spacetime, and a plausible approach to quantum gravity as a theory about quantum observables as physical quantities including the notion of quantum coordinate transformation.

Simulation of kerr nonlinearity: revealing initial state dependency

We simulate coherent driven free dissipative Kerr nonlinear system numerically using time-evolving block decimation (TEBD) algorithm and time propagation on the Heisenberg equation of motion using Euler’s method to study how the numerical results are analogous to classical bistability . The system evolves through different trajectories to stabilize different branches for different external drives and initial conditions. The Wigner state reprentation confirms the system to suffer a residual effect of initial state throughout the non-classical dynamical evolution and the metastable states of the system . Furthermore, we also see the numerically simulated spectral density remains significantly different from analytical counterparts when initial states do not lie to the same branch of the final state.

Microscopic Hamiltonian and chaotic behavior for Barium Titanate nanoparticles revealed by photonic tunneling model

In order to understand the microscopic properties of the ferroelectric nanoparticles, we review concepts of the attractive BaTiO3 crystals where a photon field from its constituent electrons and ions are considered. We present an associated Hamiltonian and extended potential which are the bases of the physical description of the system from certain geometry of the semiquantal space with the specific phonon mode. Furthermore, we formulate the nonlinear dynamics equation of the systems with the famous microscopic Hamiltonian and Heisenberg equation of motion. In addition, we interest ourselves to a system of BaTiO3 crystals and hypothesis leading to order-disorder nature of phase transitions. The latter are linked to the cooperative phenomenon of chains. The numerical simulations reveal that complex behaviors propagate in such system that can be useful for future application of Barium Titanate crystals in modern electronic technologies.

Graviton noise: The Heisenberg picture

We study the geodesic deviation equation for a quantum particle in a linearized quantum gravitational field. Particle’s Heisenberg equations of motion are treated as stochastic equations with a quantum noise. We explore the stochastic equation beyond its local approximation as a differential equation. We discuss the squeezed states resulting from an inflationary evolution. We calculate the noise in the thermal and squeezed states.

Four-wave mixing properties in bilayer graphene nanosystem

Graphene thin films are often used to manufacture various optoelectronic nanodevices owing to their advantages such as light weight, small size, high quality factor, and good conductivity. So far, there have been few studies of the four-wave mixing characteristics in a bilayer graphene nanosystem, especially theoretical research. In this work, we study theoretically the four-wave mixing properties in a bilayer graphene nanosystem. Firstly, the analytical formula of the four-wave mixing signal is derived by quantum mechanical method, which is divided into three steps. 1) Total Hamiltonian of the system is written in the rotating wave approximation. 2) By using the Heisenberg equation of motion and the commutation relations between different operators, the corresponding density matrix equations are obtained. 3) To solve these density matrix equations, we make an ansatz and obtain the analytical formula of the four-wave mixing signal. Secondly, we explore the dependence of the four-wave mixing signal on the phonon-exciton coupling strength, pumping intensity and the detuning between the exciton and the pump field. The calculated results show that the lineshape of four-wave mixing spectrum can be switched among two-peaked, three-peaked, four-peaked, five-peaked and six-peaked by adjusting the phonon-exciton coupling strength, the pumping intensity, and the detuning between the exciton and the pump field. In a weak phonon-exciton coupling regime (i.e. phonon-exciton coupling strength <i>g</i> < dephasing rate of exciton <i>Γ</i><sub>2</sub>), the intensity of the left peak and right peak of four-wave mixing signal first increase and then decrease with the increase of the pumping intensity <inline-formula><tex-math id="M1">\begin{document}$ {\varOmega }_{{\text{pu}}}^{\text{2}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230012_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230012_M1.png"/></alternatives></inline-formula>. In the intermediate and strong phonon-exciton coupling regime (i.e. <i>g</i> = <i>Γ</i><sub>2</sub> and <i>g</i> > <i>Γ</i><sub>2</sub>), the four-wave mixing spectrum exhibits a two-peaked structure. The maximum values of these two peaks increase as <inline-formula><tex-math id="M2">\begin{document}$ {\varOmega }_{{\text{pu}}}^{\text{2}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230012_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230012_M2.png"/></alternatives></inline-formula> increases, and their spacing is equal to 2<i>g</i>. Especially, for a given pumping intensity <inline-formula><tex-math id="M3">\begin{document}$ {\varOmega }_{{\text{pu}}}^{\text{2}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230012_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20230012_M3.png"/></alternatives></inline-formula> (= 10 THz<sup>2</sup>), the maximum value of the peak for <i>g</i> = 4 THz becomes 0.4% of that for <i>g</i> = 1 THz, indicating that the phonon-exciton coupling inhibits the enhancement of the four-wave mixing signal to a certain extent. Our findings can not only offer an efficient way to measure the phonon-exciton coupling strength in the bilayer graphene system, but also help one to further explore the underlying physical mechanism in such a nanosystem.

Koopman Operator and Path Integral of Quantum Free-Electron Laser Model

A quantum model of a free-electron laser (FEL) is considered. Two different approaches for the exploration of the the FEL system are considered. In the first case, the Heisenberg equations of motion are mapped on the basis of the initial wave functions, which consists of the photon coherent states and many-dimensional electron coherent states. This mapping is an exact procedure, which makes it possible to obtain an exact equation of motion for the intensity of the laser field in a closed form. The obtained equation is controlled by a Koopman operator. The analytical expression for the evolution of the FEL intensity is obtained in the framework of a perturbation theory, which is constructed for a small time scale. The second way of the consideration is based on the construction of the many-dimensional path integrals for the evolution of the wave function. This method also makes it possible to estimate the time evolution and the gain of the FEL intensity.

Spin-wave theory in a randomly disordered lattice: A Heisenberg ferromagnet

Starting from the hamiltonian for the Heisenberg ferromagnet which comprise randomly distributed nonmagnetic ions as impurities in a Bravais lattice, we express the spin operators by means of the Dyson–Maleev transformation in terms of the Bose operators of the second quantization. Then by using methods of quantum statistical field theory, we derive the partition function and the free energy for the system. We adopt the Matsubara thermal perturbation method to a portion of the hamiltonian which describes the interaction between magnons and the stationary field of nonmagnetic ions. Upon averaging over all possible distributions of impurities, we express the free energy of the system as a function of the mean impurity concentration. Subsequently, we set up the double-time single particle Green function at temperature T in the momentum space in terms of magnon operators and derive the equation of motion for the Green function through the Heisenberg equation of motion and then solve the resulting equation. From this, we calculate the self-energy and then the spectral density function for the system. We apply the formalism to the case of the simple cubic lattice and compute the density of states, the spectral density function and the lifetime of the magnons as a function of energy for several values of the mean concentration of nonmagnetic ions in the ferromagnetic lattice. We calculate the magnon energy spectrum as a function of the average impurity concentration fraction c, which shows that for low lying states, the excitation energy increases continuously with c in the studied range 0.1≤c≤0.7. We also use the spectral density function to compute some thermodynamical quantities through the magnon occupation number. We have obtained closed form expressions for the configurationally averaged physical quantities of interest in a unified fashion as functions of the mean concentration of nonmagnetic impurities c to any order of c applicable below a critical percolation concentration cp. The quantities of interest comprise the thermodynamic potential (free energy), the spin-wave self-energy and the spectral density function from which other quantities can be derived.

Tomography of time-dependent quantum Hamiltonians with machine learning

Interacting spin networks are fundamental to quantum computing. Data-based tomography of time-independent spin networks has been achieved, but an open challenge is to ascertain the structures of time-dependent spin networks using time series measurements taken locally from a small subset of the spins. Physically, the dynamical evolution of a spin network under time-dependent driving or perturbation is described by the Heisenberg equation of motion. Motivated by this basic fact, we articulate a physics-enhanced machine learning framework whose core is Heisenberg neural networks. In particular, we develop a deep learning algorithm according to some physics motivated loss function based on the Heisenberg equation, which "forces" the neural network to follow the quantum evolution of the spin variables. We demonstrate that, from local measurements, not only the local Hamiltonian can be recovered but the Hamiltonian reflecting the interacting structure of the whole system can also be faithfully reconstructed. We test our Heisenberg neural machine on spin networks of a variety of structures. In the extreme case where measurements are taken from only one spin, the achieved tomography fidelity values can reach about 90%. The developed machine learning framework is applicable to any time-dependent systems whose quantum dynamical evolution is governed by the Heisenberg equation of motion.

Two-step Dynamics of Photoinduced Phonon Entanglement Generation between Remote Electron–Phonon Systems

The generation of quantum entanglement between phonons in photoirradiated remote electron-phonon systems is numerically studied. Upon excitation by a visible/ultraviolet laser pulse, the entanglement of electrons is immediately generated and that of phonons follows via electron-phonon interactions, i.e., the entanglement generation of phonons is a two-step process. Therefore, it is important to design the temporal properties of incident optical pulses in order to control the entanglement of electrons and/or phonons. These features are revealed by the quantum mutual information and the composite modes derived from the Heisenberg equation of motion. The calculated results also show that the dynamics of the phonon entanglement can be observed by time-resolved spectroscopy on the scattered light.

Exact density matrix elements for a driven dissipative system described by a quadratic Hamiltonian

For a prototype quadratic Hamiltonian describing a driven, dissipative system, exact matrix elements of the reduced density matrix are obtained from a generating function in terms of the normal characteristic functions. The approach is based on the Heisenberg equations of motion and operator calculus. The special and limiting cases are discussed.

Resonance interaction of two entangled atoms accelerating between two mirrors

We study the resonance interaction between two entangled identical atoms coupled to a quantized scalar field vacuum, and accelerating between two mirrors. We show how radiative processes of the two-atom entangled state can be manipulated by the atomic configuration undergoing noninertial motion. Incorporating the Heisenberg picture with symmetric operator ordering, the vacuum fluctuation and the self-reaction contributions are distinguished. We evaluate the resonance energy shift and the relaxation rate of energy of the two atom system from the self-reaction contribution in the Heisenberg equation of motion. We investigate the variation of these two quantities with relevant parameters such as atomic acceleration, interatomic distance and position with respect to the boundaries. We show that both the energy level shift and the relaxation rate can be controlled by tuning the above parameters.

Anomalous Conductance in Trans-Polyacetylene Chains with Even–Odd Parity

This work presents an analytical study of electronic transport in dimerized trans-polyacetylene (Trans PA) oligomers containing even (n = 4, 6, 8, 10 sites) and odd (n = 3, 5, 7, 9 sites) chains where the site C1 is sandwiched by two metallic electrodes (Left and Right). These devices exhibit T-shaped geometry and are investigated by Su-Schrieffer-Heeger (SSH) model via Heisenberg equation of motion combined with the Keldysh formalism. We introduced disorder into the system through the dimerization force (δ) demonstrating that the dimerization in the chain can effectively lead to a linear (low δ) or zigzag (high δ) behavior, besides also suffering increase or decrease in conductance peak [dI/dV]max. The odd chains exhibit trivial topological behavior to which the conductance peaks are suppressed as dimerization disorder is considered-creates barriers for tunneling. On the other hand, we have an opposite behavior for conductance peaks in the even chains. For example, the chain with weak dimerization (low δ) has a perfect transmission for even chains. In addition, we note for odd chains the formation of a plateau in the I–V curve for bias voltages and for even chains that show a linear current. This procedure shows an analytical study through the tunneling of the parameters on the device, such parameters as the tunneling amplitude (ГL/R) can be accessible experimentally.

Quantum Kerr nonlinear coupler: analytical versus phase-space method

The generation of squeezed states of light in a two-mode Kerr nonlinear directional coupler (NLDC) was investigated using two different methods in quantum mechanics. First, the analytical method, a Heisenberg-picture-based method where the operators are evolving in time but the state vectors are time-independent. In this method, an analytical solution to the coupled Heisenberg equations of motion for the propagating modes was proposed based on the Baker–Hausdorff (BH) formula. Second, the phase space method, a Schrödinger-picture-based method in which the operators are constant and the density matrix evolves in time. In this method, the quantum mechanical master equation of the density matrix was converted to a corresponding classical Fokker–Planck (FP) equation in positive-P representation. Then, the FP equation was converted to a set of stochastic differential equations using Ito rules. The strengths and weaknesses of each method are discussed. Good agreement between both methods was achieved, especially at early evolution stages and lower values of linear coupling coefficient. On one hand, the analytical method seems insensitive to higher values of nonlinear coupling coefficients. Nevertheless, it demonstrated better numerical stability. On the other hand, the solution of the stochastic equations resulting from the phase space method is numerically expensive as it requires averaging over thousands of trajectories. Besides, numerically unstable trajectories appear with positive-P representation at higher values of nonlinearity.

Influence of the dissipation on the N-level atom interacting with a two two-level atoms in presence of qubit\u2013qubit interaction

The interacting of two qubits and an N-level atom based on su(2) Lie algebra in the presence of both qubit–qubit interaction and dissipation term is considered. The effects of the qubit–qubit interaction and the dissipation term on the dynamics of the proposed system are discussed in detail for certain values of the number of levels. The dynamical expressions of the observable operators are obtained using the Heisenberg equation of motion. The population inversion and linear entropy, as well as the concurrence formula as a measure of entanglement between the two qubits are calculated and discussed. The roles of the number of levels, the qubit–qubit coupling parameter and the dissipation rate on these quantities are also discussed. We explore the sudden birth and sudden death of the entanglement phenomena with and without the dissipation term.

The Damping and Diffusion of Atoms Moving in the Background Electromagnetic Environment

The interaction between an atom and the quantized electromagnetic field depends on the position of the atom. Then the atom experiences a force which is the minus gradient of this interaction. Through the Heisenberg equations of motion and the Born-Markov approximation, the mean and correlation of the force are obtained, showing that the center-of-mass motion of the atom is damped and diffused. This approach can be easily generalized to multi-level atoms, where the damping force and diffusion coefficients are just the weighted average of the contributions from all pairs of energy levels that have nonvanishing dipole elements. It is shown that these results are invariant under Galilean transformation, and in principle can be used to determine the velocity of the lab relative to the background radiation.

Quantum analysis of a traveling-wave electro-optic phase modulator in the presence of the phase noise from a radio frequency oscillator and a laser

Quantum analysis has been done for the traveling-wave (TW) electro-optic phase modulator in the presence of the phase noise of a radio frequency (RF) oscillator and a laser source, in which the Hamiltonian for an active section of the phase modulator system is derived. The effect of RF phase noise from the oscillator and a phase velocity mismatch (PVM) between optical photons and the RF field are included in the Hamiltonian. The time evolution of the field operator of the laser in the TW electro-optic phase modulator is derived by the Heisenberg equation of motion. The action of a TW electro-optic phase modulator in the presence of the two distinct weak noises—classical phase noise because of RF and quantum phase noise caused by a laser—has been evaluated and explained in terms of the second-order temporal correlation function and power spectrum. The power spectrum has been written as a function of the semiconductor laser linewidth, an enhancement factor in the laser linewidth, and the RF oscillator linewidth. The results are applied to analyze the basic model of a frequency-coded quantum key distribution (FC-QKD) system. The quantum bit error rate (QBER), the first-order optical intensity correlation function, and the spectrum of intensity fluctuation have been calculated for the partially coherent regime to evaluate the performance of the FC-QKD system.

Influence of Coulomb Repulsive Interaction and Image Potential on Electron Tunneling between Alkali-Metal Ions and Solid Surfaces

On the basis of numerical results obtained by solving simultaneous differential equations derived from the Heisenberg equation of motion, we examined the influence of the Coulomb repulsive interact...

Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion

In this article we model a financial derivative price as an observable on the market state function. We apply geometric techniques to integrating the Heisenberg Equation of Motion. We illustrate how the non-commutative nature of the model introduces quantum interference effects that can act as either a drag or a boost on the resulting return. The ultimate objective is to investigate the nature of quantum drift in the Accardi-Boukas quantum Black-Scholes framework which involves modelling the financial market as a quantum observable, and introduces randomness through the Hudson-Parthasarathy quantum stochastic calculus. In particular we aim to differentiate randomness that is introduced through external noise (quantum stochastic calculus) and randomness that is fundamental to a quantum system (Heisenberg Equation of Motion).

Influence of classical field on entanglement and photon statistics of n-level atom interacting with a two two-level atom

In this study, we consider three interacting atoms, one of them represented by N-level atom based on SU(2) Lie algebra and the other represented by a two two-level atom in presence of the external field. The effect of the external field on the dynamics of the proposed system is discussed in detail for certain values for the external field. The dynamical expression of the observable operators is obtained by using the Heisenberg equation of motion. The general solution via solving Schrodinger equation is obtained. The fidelity and concurrence formula as a measure of entanglement between two two-level atom are calculated and discussed in detail. We explore the sudden death and sudden birth phenomena with and without the presence of external field. Finally, we compare the results of the fidelity, concurrence and second-order correlation function for some values of the initial state and the external field parameters.

Influence of classical field on entanglement and photon statistics of n-level atom interacting with a two two-level atom

In this study, we consider three interacting atoms, one of them represented by N-level atom based on SU(2) Lie algebra and the other represented by a two two-level atom in presence of the external field. The effect of the external field on the dynamics of the proposed system is discussed in detail for certain values for the external field. The dynamical expression of the observable operators is obtained by using the Heisenberg equation of motion. The general solution via solving Schrodinger equation is obtained. The fidelity and concurrence formula as a measure of entanglement between two two-level atom are calculated and discussed in detail. We explore the sudden death and sudden birth phenomena with and without the presence of external field. Finally, we compare the results of the fidelity, concurrence and second-order correlation function for some values of the initial state and the external field parameters.