Heisenberg Representation Research Articles

Spent Time of Orbiting Eletron Generating the Quantized Energy En in the Hydrogen Like Atom

Despite the large literature on hydrogen-like atoms, one cannot find any mention as to how long an electron spends in the orbit to generate the quanized radiation energy En. Here, the Heisenberg representation of operators is exteded so as to be able to deal also with imverse oprators. This is necessary when dealing with Coulomb potentials. This then allows to write down the general hydrogen-like atom in a compact form. With the help of operator algebra that includes also the inverse operators, one arrives at the differential equation of the electron trajectory with respect to time.The time solution is achieved through established coupled integral equations. While already established quantized radiation electron energy En is proportional to 1/n2, the newely derived quantized electron circular time tn(e) is proportional to n3 with n=1,2,3,..,and tn(e) being of the order of 10−16 sec in magnitude.

Valuation of a financial claim contingent on the outcome of a quantum measurement

We consider a rational agent who at time 0 enters into a financial contract for which the payout is determined by a quantum measurement at some time T > 0. The state of the quantum system is given in the Heisenberg representation by a known density matrix p^ . How much will the agent be willing to pay at time 0 to enter into such a contract? In the case of a finite dimensional Hilbert space H , each such claim is represented by an observable X^T where the eigenvalues of X^T determine the amount paid if the corresponding outcome is obtained in the measurement. We prove, under reasonable axioms, that there exists a pricing state q^ which is equivalent to the physical state p^ such that the pricing function Π0T takes the linear form Π0T(X^T)=P0Ttr(q^X^T) for any claim X^T , where P0T is the one-period discount factor. By ‘equivalent’ we mean that p^ and q^ share the same null space: that is, for any |ξ⟩∈H one has p^|ξ⟩=0 if and only if q^|ξ⟩=0 . We introduce a class of optimization problems and solve for the optimal contract payout structure for a claim based on a given measurement. Then we consider the implications of the Kochen–Specker theorem in this setting and we look at the problem of forming portfolios of such contracts. Finally, we consider multi-period contracts.

Klein-Gordon Theory in Noncommutative Phase Space

We extend the three-dimensional noncommutative relations of the position and momentum operators to those in the four dimension. Using the Seiberg-Witten (SW) map, we give the Heisenberg representation of these noncommutative algebras and endow the noncommutative parameters associated with the Planck constant, Planck length and cosmological constant. As an analog with the electromagnetic gauge potential, the noncommutative effect can be interpreted as an effective gauge field, which depends on the Plank constant and cosmological constant. Based on these noncommutative relations, we give the Klein-Gordon (KG) equation and its corresponding current continuity equation in the noncommutative phase space including the canonical and Hamiltonian forms and their novel properties beyond the conventional KG equation. We analyze the symmetries of the KG equations and some observables such as velocity and force of free particles in the noncommutative phase space. We give the perturbation solution of the KG equation.

Nonequilibrium spectral moment sum rules of the Holstein–Hubbard model

We derive a general procedure for evaluating the nth derivative of a time-dependent operator in the Heisenberg representation and employ this approach to calculate the zeroth to third spectral moment sum rules of the retarded electronic Green’s function and self-energy for a system described by the Holstein–Hubbard model allowing for arbitrary spatial and time variation of all parameters (including spatially homogeneous electric fields and parameter quenches). For a translationally invariant (but time-dependent) Hamiltonian, we also provide sum rules in momentum space. The sum rules can be applied to various different phenomena like time-resolved angle-resolved photoemission spectroscopy and benchmarking the accuracy of numerical many-body calculations. This work also corrects some errors found in earlier work on simpler models.

Quantum tomography of entangled qubits by time-resolved single-photon counting with time-continuous measurements

In this article, we introduce a framework for entanglement characterization by time-resolved single-photon counting with measurement operators defined in the time domain. For a quantum system with unitary dynamics, we generate time-continuous measurements by shifting from the Schrödinger picture to the Heisenberg representation. In particular, we discuss this approach in reference to photonic tomography. To make the measurement scheme realistic, we impose timing uncertainty on photon counts along with the Poisson noise. Then, the framework is tested numerically on quantum tomography of qubits. Next, we investigate the accuracy of the model for polarization-entangled photon pairs. Entanglement detection and precision of state reconstruction are quantified by figures of merit and presented on graphs versus the amount of time uncertainty.

Time Evolution of the Position Operators in a Bilayer Graphene

In this work, the researchers mainly focus on the trembling motion which is known as Zitterbewegung in a bilayer grapheme. This is effectively achieved by means of the long-wave approximation. That is, the Heisenberg representation is ultimately employed in order to derive the analytical expression concerning the expectation value related to the position operator along the longitudinal and transversal orientation, which describes the motion concerning the electronic wave packet inside the bilayer graphene. Parameters’ numbers are considered to explicate the packet of Gaussian wave, including the polarization of initial pseudo-spin as well as the wave number of the initial carrier number along with the localized wave packet’s width along the longitudinal as well as transversal orientation. Consequently, the researchers show that the obvious oscillation in position operator can be effectively controlled not only by what is known as the initial parameters concerning the wave packet. Rather, it can mainly be controlled by selecting the localized quantum state’s components. Furthermore, the interference’s analysis between the conduction as well as valence bands concerning quantum states is really emphasized as the ability of what can be described as the transient’s emergence, or in a sense, aperiodic temporal oscillations concerning the average value of position operator in the bilayer graphene.

Heisenberg representation of nonthermal ultrafast laser excitation of magnetic precessions

We derive the Heisenberg representation of the ultrafast inverse Faraday effect that provides the time evolution of magnetic vectors of a magnetic system during its interaction with a laser pulse. We obtain a time-dependent effective magnetic operator acting in the Hilbert space of the total angular momentum that describes a process of nonthermal excitation of magnetic precessions in an electronic system by a circularly polarized laser pulse. The magnetic operator separates the effect of the laser pulse on the magnetic system from other magnetic interactions. The effective magnetic operator provides the equations of motion of magnetic vectors during the excitation by the laser. We show that magnetization dynamics calculated with these equations is equivalent to magnetization dynamics calculated with the time-dependent Schr\"odinger equation, which takes into account the interaction of an electronic system with the electric field of light. We model and compare laser-induced precessions of magnetic sublattices of an easy-plane and an easy-axis antiferromagnetic systems. Using these models, we show how the ultrafast inverse Faraday effect induces a net magnetic moment in antiferromagnets and demonstrate that a crystal field environment and the exchange interaction play essential roles for laser-induced magnetization dynamics even during the action of a pump pulse. Using our approach, we show that light-induced precessions can start even during the action of the pump pulse with a duration several tens times shorter than the period of induced precessions and affect the position of magnetic vectors after the action of the pump pulse.

Gottesman Types for Quantum Programs

The Heisenberg representation of quantum operators provides a powerful technique for reasoning about quantum circuits, albeit those restricted to the common (non-universal) Clifford set H, S and CNOT. The Gottesman-Knill theorem showed that we can use this representation to efficiently simulate Clifford circuits. We show that Gottesman's semantics for quantum programs can be treated as a type system, allowing us to efficiently characterize a common subset of quantum programs. We also show that it can be extended beyond the Clifford set to partially characterize a broad range of programs. We apply these types to reason about separable states and the superdense coding algorithm.

SU(1,1) interferometry with parity measurement

We present a new operator method in the Heisenberg representation to obtain the signal of parity measurement within a lossless SU(1,1) interferometer. Based on this method, it is convenient to derive the parity signal directly in terms of input states, including general Gaussian and non-Gaussian states. As applications, we revisit the signal of parity measurement within an SU(1,1) interferometer when a coherent or thermal state and a squeezed vacuum state are considered as input states. In addition, we obtain the parity signal of an arbitrary single-mode state when it passes through an SU(1,1) interferometer, which is also a new result. Then, we analytically prove that parity measurement can saturate the quantum Cramér–Rao bound when the estimated phase approaches zero. Therefore, the operator method proposed in this work may bring convenience to the study of quantum metrology, particularly the phase estimation based on an SU(1,1) interferometer.

Convergence theorems on multi-dimensional homogeneous quantum walks

We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. We prove that every homogeneous walks with finite degree of freedom have limit distribution. This theorem can also be applied to every crystal lattice. In this theorem, it is not necessary to assume that the support of the initial unit vector is finite. We also pay attention on 1-cocycles, which is related to Heisenberg representation of time evolution of observables. For homogeneous walks with finite degree of freedom, convergence of averages of 1-cocycles associated with the position observable is also proved.

Local Root Numbers for Heisenberg-Representations — Some Explicit Results

Heisenberg representations [Formula: see text] of (pro-)finite groups [Formula: see text] are by definition irreducible representations of the two-step nilpotent factor group [Formula: see text] Better known are Heisenberg groups which can be understood as allowing faithful Heisenberg representations. A special feature is that [Formula: see text] will be induced by characters [Formula: see text] of subgroups in multiple ways, where the pairs [Formula: see text] can be interpreted as maximal isotropic pairs. If [Formula: see text] is a [Formula: see text]-adic number field and [Formula: see text] the absolute Galois group then maximal isotropic pairs rewrite as [Formula: see text] where [Formula: see text] is an abelian extension and [Formula: see text] a character. We will consider the extended local Artin-root-number [Formula: see text] for those [Formula: see text] which are essentially tame and express it by a formula not depending on the various maximal isotropic pairs [Formula: see text] for [Formula: see text]

Dynamic State Reconstruction of Quantum Systems Subject to Pure Decoherence

The article introduces efficient quantum state tomography schemes for qutrits and entangled qubits subject to pure decoherence. We implement the dynamic state reconstruction method for open systems sent through phase-damping channels, which was proposed in: Czerwinski and Jamiolkowski Open Syst. Inf. Dyn. 23, 1650019 (2016). In the present article we prove that two distinct observables measured at four different time instants suffice to reconstruct the initial density matrix of a qutrit with evolution given by a phase-damping channel. Furthermore, we generalize the approach in order to determine criteria for quantum tomography of entangled qubits. Finally, we prove two universal theorems concerning the number of observables required for quantum state tomography of qudits subject to pure decoherence. We believe that dynamic state reconstruction schemes bring advancement and novelty to quantum tomography since they utilize the Heisenberg representation and allow to define the measurements in time domain.

Equilibrium properties and decoherence of an open harmonic oscillator

The equilibrium properties of an open harmonic oscillator are considered in three steps: first the creation and destruction operators are generalized for open dynamics and the creation operator is used to construct coherent states. The second step consists of the introduction of the Heisenberg representation where the dynamical decoherence is identified. Finally it is pointed out that the quantum fluctuations generate non-continuous limit for infinitesimal system–environment interactions and at the border of the under- and over-damped oscillator.

Hybrid quantum-classical simulation of quantum speed limits in open quantum systems

The quantum speed limit (QSL) provides a fundamental upper bound on the speed of quantum evolution, but its evaluation in generic open quantum systems still presents a formidable computational challenge. Herein, we introduce a hybrid quantum-classical method for computing QSL times in multi-level open quantum systems. The method is based on a mixed Wigner–Heisenberg representation of the composite quantum dynamics, in which the open subsystem of interest is treated quantum mechanically and the bath is treated in a classical-like fashion. By solving a set of coupled first-order deterministic differential equations for the quantum and classical degrees of freedom, one can compute the QSL time. To demonstrate the utility of the method, we study the unbiased spin-boson model and provide a detailed analysis of the effect of the subsystem-bath coupling strength and bath temperature on the QSL time. In particular, we find a turnover of the QSL time in the strong coupling regime, which is indicative of a speed-up in the quantum evolution. We also apply the method to the Fenna–Matthews–Olson complex model and identify a potential connection between the QSL time and the efficiency of the excitation energy transfer at different temperatures.

Contact Interactions and Gamma Convergence

In Classical Mechanics constraints describe forces restricting the motion of two systems when they are in In Quantum Mechanics in the Schrodinger representation the state of the system is described by a (probability) wave and the Schrodinger equation is dispersive (it does not preserve locality of the wave function): at any time the wave function is spread out over all space. It is difficult to define contact. To avoid this difficulty it is convenient to use the Heisenberg representation and describe the system by means of self-adjoint operators on some function space. Each self-adjoint operator operator has a domain of definition. We consider first in some detail the dynamics in R^3 and later consider the case of dimension two and dimension one. In three dimension contact interactions gives the Efimov spectrum of trimers and quadrimers in low energy physics and the Bose-Einstein condensate, both in low density and in high density. The case of dimension one is particularly interesting because the system may represent three particles on a Y-shaped graph with contact interaction at the vertex. We consider both the case of Bose particles which satisfy the Schrodinger equation and the case of spin 1\2 fermions which satisfy the Pauli equation. They form respectively Bose crystals and Fermi crystals (not necessary periodic). In both cases there is only one bound state at each vertex. In an extended crystal, due to Fermi-Dirac statistics, all bound states may be occupied to form the Fermi sea. The particles on the surface have a Dirac spectrum. We prove that in the semiclassical limit the dynamics of particles on the surface of the Fermi sea is the (classical) motion studied in detail by Novikov and Maltsev. In this semiclassical description the Fermi surface can be deformed by a magnetic field and one may have topological resonances.

Trembling motion of the wave packet in armchair graphene nanoribbons (AGNRs)

A treatment of the trembling motion or Zitterbewegung (ZB) phenomena in the armchair graphene nanoribbons (AGNRs) by using long-wave approximation is presented theoretically. We are first interested to study the time dependence of the average values for the current density in AGNR. The longitudinal and transversal components of the current density operator are derived analytically in the Heisenberg representation. The wave packet in a Gaussian distribution is considered with half-width d and a carrier wave vector in the longitudinal orientation k[Formula: see text]. The average values of the current density which represent the current induced by the motion of the electrons along the nanoribbon are calculated numerically. The interference between two energy branches, or the corresponding upper and lower energy states, leads to the trembling motion in the armchair graphene nanoribbon, and hence, our results emphasized that the phenomena of ZB has an aperiodic and nontransient oscillation in the longitudinal and transversal direction, respectively. The average values for the current density for AGNRs are calculated with extremely large value of N and compared with infinite pristine graphene.

Fractional evolution in quantum mechanics

Fractional time (non-unitary) quantum mechanics is discussed for both Schrödinger and Heisenberg representations of quantum mechanics. A correct form of the fractional Schrödinger equation is elucidated. A generic regularization procedure for the fractional Heisenberg equations of motion is suggested that makes it possible to solve these nonlinear equations in the framework of photonic and spin coherent states consideration.

Current Density and Zitterbewegung in a Monolayer Graphene

The trembling motion as known the Zitterbewegung (ZB) in the monolayer graphene is studied within the long-wave approximation. We derive an analytical expression for the operator of the current density by using the Heisenberg representation, the current density describes the current induced by the motion of the electronic wave packet on the surface of the graphene sheet. It has been taking into consideration many values for the initial pseudo-spin polarization for the wave packet. The Heisenberg representation is used to analyze the interference between the conduction and valence bands of a quantum state. The interference leads to temporal oscillations for the average values of current density in the monolayer graphene.

An FPGA-based quantum circuit emulation framework using heisenberg representation

While physical realization of practical large-scale quantum computers is still ongoing, theoretical research of quantum computing applications is facilitated on classical computing platforms through simulation and emulation methods. Nevertheless, the exponential increase in resource requirement with the increase in the number of qubits is an inherent issue in classical modeling of quantum systems. In the effort to alleviate the critical scalability issue in existing FPGA emulation works, a novel FPGA-based quantum circuit emulation framework based on Heisenberg representation is proposed in this paper. Unlike previous works that are restricted to the emulations of quantum circuits of small qubit sizes, the proposed FPGA emulation framework can scale-up to 120-qubit on Altera Stratix IV FPGA for the stabilizer circuit case study while providing notable speed-up over the equivalent simulation model.

Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics

An introduction to the use of projection-operator methods for the derivation of classical fluid transport equations for weakly coupled, magnetised, multispecies plasmas is given. In the present work, linear response (small perturbations from an absolute Maxwellian) is addressed. In the Schrödinger representation, projection onto the hydrodynamic subspace leads to the conventional linearized Braginskii fluid equations when one restricts attention to fluxes of first order in the gradients, while the orthogonal projection leads to an alternative derivation of the Braginskii correction equations for the non-hydrodynamic part of the one-particle distribution function. The projection-operator approach provides an appealingly intuitive way of discussing the derivation of transport equations and interpreting the significance of the various parts of the perturbed distribution function; it is also technically more concise. A special case of the Weinhold metric is used to provide a covariant representation of the formalism; this allows a succinct demonstration of the Onsager symmetries for classical transport. The Heisenberg representation is used to derive a generalized Langevin system whose mean recovers the linearized Braginskii equations but that also includes fluctuating forces. Transport coefficients are simply related to the two-time correlation functions of those forces, and physical pictures of the various transport processes are naturally couched in terms of them. A number of appendices review the traditional Chapman–Enskog procedure; record some properties of the linearized Landau collision operator; discuss the covariant representation of the hydrodynamic projection; provide an example of the calculation of some transport effects; describe the decomposition of the stress tensor for magnetised plasma; introduce the linear eigenmodes of the Braginskii equations; and, with the aid of several examples, mention some caveats for the use of projection operators.