Quantum Dynamical Equation Research Articles

Quantum Thermal Amplifiers with Engineered Dissipation.

A three-terminal device, able to control the heat currents flowing through it, is known as a quantum thermal transistor whenever it amplifies two output currents as a response to the external source acting on its third terminal. Several efforts have been proposed in the direction of addressing different engineering options of the configuration of the system. Here, we adhere to the scheme in which such a device is implemented as a three-qubit system that interacts with three separate thermal baths. However, another interesting direction is how to engineer the thermal reservoirs to magnify the current amplification. Here, we derive a quantum dynamical equation for the evolution of the system to study the role of distinct dissipative thermal noises. We compare the amplification gain in different configurations and analyze the role of the correlations in a system exhibiting the thermal transistor effect, via measures borrowed from the quantum information theory.

Implications of causality for quantum biology – I: topology change

ABSTRACTA framework for describing the causal, topology changing, evolution of interacting biomolecules is developed. The quantum dynamical manifold equations (QDMEs) derived from this framework can be related to the causality restrictions implied by a finite speed of light and to Planck's constant to set a transition frequency scale. The QDMEs imply conserved stress-energy, angular-momentum and Noether currents. The functional whose extremisation leads to this result provides a causal, time-dependent, non-equilibrium generalisation of the Hohenberg–Kohn theorem. The system of dynamical equations derived from this functional and the currents J derived from the QDMEs are shown to be causal and consistent with the first and second laws of thermodynamics. This has the potential of allowing living systems to be quantum mechanically distinguished from non-living ones.

Quasimap Counts and Bethe Eigenfunctions

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties. This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik-Zamolodchikov and dynamical q-difference equations.

Derivation and application of quantum Hamilton equations of motion

Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image

Quasimap Counts and Bethe Eigenfunctions

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties. This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik-Zamolodchikov and dynamical q-difference equations.

Quantum Yang-Baxter Equation, Braided Semigroups, and Dynamical Yang-Baxter Maps

We prove that braided semigroups with suitable conditions can produce solutions to the quantum Yang-Baxter equation in every tensor category. As an application, some dynamical Yang-Baxter maps, set-theoretic solutions to a version of the quantum dynamical Yang-Baxter equation, are constructed.

Universal vertex-IRF transformation for quantum affine algebras

We construct a universal solution of the generalized coboundary equation in the case of quantum affine algebras, which is an extension of our previous work to $U_q(A^{(1)}_r)$Uq(Ar(1)). This universal solution has a simple Gauss decomposition which is constructed using Sevostyanov's characters of twisted quantum Borel algebras. We show that in the evaluation representations it gives a vertex-face transformation between a vertex type solution and a face type solution of the quantum dynamical Yang-Baxter equation. In particular, in the evaluation representation of $U_q(A_1^{(1)})$Uq(A1(1)), it gives Baxter's well-known transformation between the 8-vertex model and the interaction-round-faces (IRF) height model.

Jensen–Shannon divergence and non-linear quantum dynamics

Using the statistical inference method, a non-relativistic, spinless, non-linear quantum dynamical equation is derived with the Fisher information metric substituted by the Jensen–Shannon distance information. Among all possible implications, it is shown that the non-linear Schrödinger equation preserves the symplectic structure of the complex Hilbert space, hence a Hamiltonian dynamics. The canonically projected dynamics is obtained on the corresponding projective Hilbert space of pure state density operators.

The effect of phonon modes on the D2(v=0, j=0)–Cu(111) scattering processes

We include the phonon modes originating from the three layers of Cu(111) surface atoms on the dynamics of incoming molecular [D2(v, j)] degrees of freedom (DOFs) through a mean-field approach, where the surface temperature is incorporated into the effective potential by considering the Bose–Einstein probability factor for the initial state distribution of the surface modes calculated within the harmonic approximation. Such a time- and temperature-dependent effective Hamiltonian is further subdivided assuming a weak coupling between two sets of molecular DOFs, namely (x, y, z, Z) and (X, Y), respectively, in particular, to reduce the computational cost, and the corresponding coupled quantum dynamical equations of motion have been formulated in terms of the time-dependent discrete variable representation (TDDVR) approach. We demonstrate the applicability of the TDDVR method to investigate the collision of H2(v, j) on the Cu(100) surface by calculating the reaction probabilities and scattering cross-sections. Calculated results for the D2(v=0, j=0)–Cu(111) system show that the phonon modes affect the state-to-state transition probabilities of the scattered D2 molecule substantially and chemisorption–physisorption processes noticeably.

Dynamical Yang-Baxter Maps Associated with Homogeneous Pre-Systems

We construct dynamical Yang-Baxter maps, which are set-theoretical solutions to a version of the quantum dynamical Yang-Baxter equation, by means of homogeneous pre-systems, that is, ternary systems encoded in the reductive homogeneous space satisfying suitable conditions. Moreover, a characterization of these dynamical Yang- Baxter maps is presented.

The effect of phonon modes on the H2(v, j)/D2(v, j)–Cu(1nn) scattering processes

We include the effect of the phonon modes originating from the three layers of Cu(1nn) surface atoms on the dynamics of incoming molecular [H(2)(v, j)/D(2)(v, j)] degrees of freedom (DOFs) through a mean-field approach, where the surface temperature is incorporated into the effective potential by considering Bose-Einstein probability (BEP) factor for the initial state distribution of the surface modes calculated within harmonic approximation. Such time and temperature dependent effective Hamiltonian is further subdivided assuming a weak coupling between the two sets of molecular DOFs, namely, (x, y, z, Z) and (X, Y), respectively, in particular, to reduce the computational cost and the corresponding coupled quantum dynamical equations of motion have been formulated in terms of Time Dependent Discrete Variable Representation (TDDVR) approach. We demonstrate the workability of TDDVR method to investigate the scattering of H(2)(v, j) on Cu(1nn) surface by calculating the reaction probabilities and scattering cross-sections. Calculated results show that the phonon modes affect (a) the state-to-state transition probabilities of the scattered H(2) molecule substantially but chemisorption and physisorption processes negligibly and (b) the reaction probability of the incoming D(2) molecule noticeably.

From qubits and actions to the Pauli–Schrödinger equation

Here I show that a classical or quantum bit state plus one simple operation, an action, are sufficient ingredients to derive a quantum dynamical equation that rules the sequential changes of the state. Then, by assuming that a freely moving massive particle is the qubit carrier, it is found that both, the particle position in physical space and the qubit state, change in time according to the Pauli–Schrödinger equation. So, this approach suggests the following conjecture: because it carries one qubit of information the particle motion has its description enslaved by the very existence of the internal degree of freedom. It is compelled to be described no more classically but by a wavefunction. I also briefly discuss the Dirac equation in terms of qubits.

The EllipticGL(n)Dynamical Quantum Group as an𝔥-Hopf Algebroid

Using the language of𝔥-Hopf algebroids which was introduced by Etingof and Varchenko, we construct a dynamical quantum group,ℱell(GL(n)), from the elliptic solution of the quantum dynamical Yang-Baxter equation with spectral parameter associated to the Lie algebra𝔰𝔩n. We apply the generalized FRST construction and obtain an𝔥-bialgebroidℱell(M(n)). Natural analogs of the exterior algebra and their matrix elements, elliptic minors, are defined and studied. We show how to use the cobraiding to prove that the elliptic determinant is central. Localizing at this determinant and constructing an antipode we obtain the𝔥-Hopf algebroidℱell(GL(n)).

Entangled trajectory dynamics in the Husimi representation

We solve quantum dynamical equations of simple systems by propagating ensembles of interacting trajectories. A scheme is proposed which uses adaptive kernel density estimation for representing probability distribution functions and their derivatives. The formulation is carried on in the Husimi representation to ensure the positiveness of the distribution functions. By comparing to previous work, the effect of changing representations is studied as well as the advantage of using adaptive kernels for the estimation of probability distributions. We found significant improvement in the accuracy of the results.

Dynamical Yang-Baxter maps

We propose and investigate dynamical Yang-Baxter maps, some of which produce solutions to the quantum dynamical Yang-Baxter equation. Suppose that L is a loop and a group. If their unit elements coincide, then L gives birth to a bijective dynamical Yang-Baxter map from L × L to L × L whose dynamical parameter belongs to L. The above group L is abelian if and only if the corresponding dynamical Yang-Baxter map satisfies the unitary condition.

Entanglement of an exciton in a quantum dot molecule in radiation pulses

We theoretically analyze the dynamical behaviors of an interacting electron–hole pair in a double coupled quantum dot molecule in radiation pulses. It is found the maximally entangled Bell states can be prepared when a resonant transfer occurs between two localized states. The strength and duration of selective light pulses for producing maximally entangled states are identified by analytic and exact numerical solution of the quantum dynamical equations under the stronger Coulomb interaction.

Entanglement of an exciton in a quantum dot molecule in radiation pulses

We theoretically analyze the dynamical behaviors of an interacting electron–hole pair in a double coupled quantum dot molecule in radiation pulses. It is found the maximally entangled Bell states can be prepared when a resonant transfer occurs between two localized states. The strength and duration of selective light pulses for producing maximally entangled states are identified by analytic and exact numerical solution of the quantum dynamical equations under the stronger Coulomb interaction.

Quantum dynamics of polaron formation with the Wigner-function approach

In compound semiconductor crystals with a substantial degree of ionicity Froelich interaction of charge carriers with longitudinal optical (LO) phonons is strong enough to produce detectable polaron effects. In this paper polaron effects are analysed using a quantum theory of electron transport based on the momentum- and frequency-dependent Wigner function fw(p, ω), defined starting from the G< Green function, simply related to the electron spectral function A(p, ω). The theoretical approach considers the dynamical evolution of the electron Wigner function in the presence of phonon scattering. An elaboration of the quantum dynamical equation in terms of Wigner paths formed by free flights and scattering events is used. These paths are especially suitable for a Monte Carlo solution of the transport equation for the Wigner function very similar to the semiclassical traditional Monte Carlo solution of the Boltzmann equation. Numerical results for GaAs and CdTe in a variety of physical conditions are presented.

Quantum dynamical manifolds: Pair states in high‐temperature superconductivity

AbstractThe formation of Dirac pair states is shown to be crucial to the formation of the superconductive state. The properties of these Dirac bipolaron states is studied using a quantum dynamical extension of Cartan's moving frame approach to differential geometry. The overall geometric method, called quantum dynamical manifold theory (QDMT), is used to obtain the Dirac quasiparticle pair states (cooperons and potential cooperons or Dirac polaron pairs) in a polarizable medium appropriate for high‐temperature superconductors (HTSCs). The matrix solutions of the equations describing the quantum dynamical manifold have a (flavor) symmetry also occurring in other closed‐shell systems. By independently coupling each Dirac electron to a polarizable background in a semiclassic approximation, and solving the resulting quantum dynamical manifold equations (QDMEs), there are found new, nonperturbative, analytic solutions to the polaron pair (bipolaron) problem. This enables us to show that these cooperon states are closed shells with su(2) Lie algebraic symmetry. These are used to describe the cooperon system above and below the superconduction critical temperature. It is shown how these Dirac polaron pairs can be correlated for electron–electron interactions via virtual exciton exchange and screened Coulomb repulsion and then used in a homogeneous Bethe–Salpeter equation approach to study the onset of superconductivity of polaronic superconductors involving pairs of Dirac polarons. The relation of these states to the formation of a Bose–Einstein condensation (BEC) is discussed. To numerically investigate the question of the existence of high‐temperature superconductivity mediated by the screened Coulomb and boson exchange of virtual excitons, a wave vector‐ and frequency‐dependent effective interaction, Veff(k, ω), between electron or hole polaron pairs in the electronic polaron model is used in lieu of a full pairing interaction, thereby simplifying the analysis of interacting pair states. We include the temperature dependence of the dielectric function, ε−1(k, ω, T), for many‐body systems of these quasiparticles. An effective mass method is employed to further simplify the computations. Analysis of the results shows that the attractive part of Veff for the electronic polaron model persists to high temperature. Further, polarons not participating in the superconductive state produce a measurable Fermi surface even in the presence of the bipolaron BEC. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Quantum dynamics of polaron formation with the Wigner-function approach

In compound semiconductor crystals with a substantial degree of ionicity Froelich interaction of charge carriers with longitudinal optical phonons is strong enough to produce detectable polaron effects. In this paper polaron effects are analyzed using a quantum theory of electron transport based on the momentum and frequency-dependent Wigner function ${f}_{w}(p,\ensuremath{\omega}),$ defined starting from the ${G}^{l}$ Green function, simply related to the electron spectral function $A(p,\ensuremath{\omega}).$ The theoretical approach considers the dynamical evolution of the electron Wigner function in presence of phonon scattering. An elaboration of the quantum dynamical equation in terms of Wigner paths formed by free flights and scattering events is used. These paths are especially suitable for a Monte Carlo solution of the transport equation for the Wigner function very similar to the semiclassical traditional Monte Carlo solution of the Boltzmann equation. Numerical results for GaAs and CdTe in a variety of physical conditions are presented.