Dependent Hamiltonian Research Articles

Unstructured Adiabatic Quantum Search

In the adiabatic quantum computation model, a computational procedure is described by the continuous time evolution of a time dependent Hamiltonian. We apply this method to the Grover's problem, i.e., searching a marked item in an unstructured database. Classically, the problem can be solved only in a running time of order O(N) (where N is the number of items in the database), whereas in the quantum model a speed up of order $$O(\sqrt {N)} $$ has been obtained. We show that in the adiabatic quantum model, by a suitable choice of the time-dependent Hamiltonian, it is possible to do the calculation in constant time, independent of the the number of items in the database. However, in this case the initial time-complexity of $$O(\sqrt {N)} $$ is replaced by the complexity of implementing the driving Hamiltonian.

Novel approach to the study of quantum effects in the early Universe

We develop a theoretical frame for the study of classical and quantum gravitational waves based on the properties of a nonlinear ordinary differential equation for a function $\sigma(\eta)$ of the conformal time $\eta$, called the auxiliary field equation. At the classical level, $\sigma(\eta)$ can be expressed by means of two independent solutions of the ''master equation'' to which the perturbed Einstein equations for the gravitational waves can be reduced. At the quantum level, all the significant physical quantities can be formulated using Bogolubov transformations and the operator quadratic Hamiltonian corresponding to the classical version of a damped parametrically excited oscillator where the varying mass is replaced by the square cosmological scale factor $a^{2}(\eta)$. A quantum approach to the generation of gravitational waves is proposed on the grounds of the previous $\eta-$dependent Hamiltonian. An estimate in terms of $\sigma(\eta)$ and $a(\eta)$ of the destruction of quantum coherence due to the gravitational evolution and an exact expression for the phase of a gravitational wave corresponding to any value of $\eta$ are also obtained. We conclude by discussing a few applications to quasi-de Sitter and standard de Sitter scenarios.

Mean field theories for the description of diffusion and phase transformations controlled by diffusion

A self-consistent mean field (SCMF) theory starting from an atomic diffusion model can predict most of the correlation effects induced by the vacancy diffusion mechanism while being coherent with thermodynamics. For that purpose a time dependent effective Hamiltonian is introduced into the non-equilibrium distribution function. Such effective Hamiltonian makes a new hierarchy of approximations to appear in parallel to the thermodynamic mean-field approximations. We explain how each level of approximation is related to correlation effects and how kinetics of phase transformation results from both thermodynamics and diffusion properties. A first example is the radiation induced segregation (RIS) at grain boundary (GB) in austenitic steels where thermodynamics is shown to play an important role under irradiation. In a second example, we demonstrate how mean-field approximations combined with mesoscopic theories like the classical nucleation theory can describe the kinetics of precipitation, leading to a prediction of the cluster size distributions in the solid solution and of the steady state nucleation rates in agreement with kinetic Monte Carlo simulations.

Optimal control theory for unitary transformations

The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time dependent Hamiltonian. The inverse problem of finding the field that generates a specific unitary transformation is the subject of study. The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space. Optimal control theory (OCT) is used to solve the inversion problem irrespective of the initial input state. A unified formalism, based on the Krotov method is developed leading to a new scheme. The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X^1\Sigma^+_g$ electronic state of Na$_2$. Raman-like transitions through the $A^1\Sigma^+_u$ electronic state induce the transitions. Light fields are found that are able to implement the Fourier transform within a picosecond time scale. Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse. Out of the schemes studied the square modulus scheme converges fastest. A study of the implementation of the $Q$ qubit Fourier transform in the Na$_2$ molecule was carried out for up to 5 qubits. The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels. The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized.

Energy and efficiency of adiabatic quantum search algorithms

We present the results of a detailed analysis of a general, unstructured adiabatic quantum search of a data base of $N$ items. In particular we examine the effects on the computation time of adding energy to the system. We find that by increasing the lowest eigenvalue of the time dependent Hamiltonian {\it temporarily} to a maximum of $\propto \sqrt{N}$, it is possible to do the calculation in constant time. This leads us to derive the general theorem which provides the adiabatic analogue of the $\sqrt{N}$ bound of conventional quantum searches. The result suggests that the action associated with the oracle term in the time dependent Hamiltonian is a direct measure of the resources required by the adiabatic quantum search.

Dependence of energy gap on magnetic field in semiconductor nano-scale quantum rings

We study the electron and hole energy states for a complete three-dimensional (3D) model of semiconductor nano-scale quantum rings in an external magnetic field. In this study, the model formulation includes: (i) the position dependent effective mass Hamiltonian in non-parabolic approximation for electrons, (ii) the position dependent effective mass Hamiltonian in parabolic approximation for holes, (iii) the finite hard wall confinement potential, and (iv) the Ben Daniel–Duke boundary conditions. To solve this 3D non-linear problem, we apply the non-linear iterative method to obtain self-consistent solutions. We find a non-periodical oscillation of the energy band gap between the lowest electron and hole states as a function of external magnetic fields. The result is useful in describing magneto-optical properties of the nano-scale quantum rings.

QUANTUM TREATMENT OF TIME DEPENDENT COUPLED OSCILLATORS

In this paper we consider the most quadratic time dependent Hamiltonian. An exact solution of the wave function in both the Schrödinger picture and coherent states representation is given. Linear and quadratic invariants are discussed. The eigenvalues and the corresponding eigenfunctions are obtained. The expectation values for the energy are also given.

Adiabatic quantum computation and Deutsch’s algorithm

We show that by a suitable choice of a time dependent Hamiltonian, Deutsch's algorithm can be implemented by an adiabatic quantum computer. We extend our analysis to the Deutsch-Jozsa problem and estimate the required running time for both global and local adiabatic evolutions.

A High-Pressure Spectral Hole Burning Study of Correlation between Energy Disorder and Excitonic Couplings in the LH 2 Complex from Rhodopseudomonas Acidophila

Low-temperature (5 K) nonphotochemical hole burning and absorption spectroscopies were used to study the effects of high pressure (up to 1015 MPa) on excitonic coupling energies and energy disorder in the light-harvesting complex 2 (LH 2) from Rhodopseudomonas Acidophila (strain 10050). The pressure dependences of the widths and positions of the B850 and B870 (lowest exciton level of the B850 ring) bands were determined. The results show, for the first time, that these parameters are linearly correlated (common scaling law), consistent with recent theoretical predictions (Jang; et al. J. Phys. Chem. B 2001, 105, 6655). Excitonic calculations for the B850 ring of bacteriochlorophyll a molecules were performed in the nearest dimer−dimer coupling approximation (Wu; et al. J. Phys. Chem. B 1997, 101, 7654) with E1+ diagonal energy disorder. Such a symmetry is known to dictate the response of the B870 level (absorption bandwidth, displacement (ΔE) from the B850 band maximum) to either diagonal or off-diagonal energy disorder. The results reveal that the pressure dependences of nearest neighbor coupling energies and the width of the distribution function for energy disorder are positively correlated, both increasing with pressure. Importantly, an effective pressure dependent excitonic Hamiltonian for the B850 ring was obtained. This Hamiltonian was used to show that mixing of the upper excitonic levels of the B850 ring in close proximity to the B800 levels is too weak to account for the additional (other than B800 → B850 energy transfer) decay channel of the B800 band. A simple combinatorial model for intra-B800 band downward energy transfer is presented that qualitatively explains all available low-temperature data on the additional decay channel.

Inhomogeneous quasi-stationary states in a mean-field model with repulsive cosine interactions

The system of N particles moving on a circle and interacting via a global repulsive cosine interaction is well known to display spatially inhomogeneous structures of extraordinary stability starting from certain low energy initial conditions. The object of this paper is to show in a detailed manner how these structures arise and to explain their stability. By a convenient canonical transformation we rewrite the Hamiltonian in such a way that fast and slow variables are singled out and the canonical coordinates of a collective mode are naturally introduced. If, initially, enough energy is put in this mode, its decay can be extremely slow. However, both analytical arguments and numerical simulations suggest that these structures eventually decay to the spatially uniform equilibrium state, although this can happen on impressively long time scales. Finally, we heuristically introduce a one-particle time dependent Hamiltonian that well reproduces most of the observed phenomenology.

Memory kernels and effective Hamiltonians from time-dependent methods. II. Vibrational predissociation

The overlapping resonance regime is studied from a temporal viewpoint for a model vibrational predissociation in the framework of the Feshbach partitioning method. The memory kernels that are related by Fourier transformation to the energy dependent effective Hamiltonians are computed by wave packet propagation. The characteristic time τmemory of the memory kernel is compared with the time scale of the dynamics in the bound subspace. An approximate expression for τmemory is derived. τmemory depends on the bound-free couplings on a wide energy range. It is shown that the nonoscillatory shape of these couplings as a function of the dissociation energy is a typical feature of the vibrational predissociation.

BLOCH DYNAMICS IN SPATIALLY LOCAL INHOMOGENEOUS ELECTRIC FIELDS

The influence of local inhomogeneities on the electric field dependent properties of Bloch electrons is studied. The homogeneous electric field is described through the use of the vector potential, and the instantaneous Wannier functions of the homogeneous field dependent Hamiltonian are used as bases states to depict Bloch dynamics and properties. Model examples are treated using Slater-Koster inhomogeneities and nearest-neighbor tight-binding band structure in a one dimensional, single-band analysis. Detailed analysis is presented for the special case of a constant electric field; here the influence of localization due to the presence of the electric field is shown to clearly affect the energy spectrum of the Bloch electron for a single and double impurity configuration.

Parametric dependent Hamiltonians, wave functions, random matrix theory, and quantal-classical correspondence.

We study a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where (Q,P) are the canonical coordinates of a particle in a two-dimensional well, and x is a parameter. By changing x we can deform the "shape" of the well. The quantum eigenstates of the system are /n(x)>. We analyze numerically how the parametric kernel P(n/m)=/<n(x)/m(x(0))>/(2) evolves as a function of delta(x)[triple bond](x-x(0)). This kernel, regarded as a function of n-m, characterizes the shape of the wave functions, and it also can be interpreted as the local density of states. The kernel P(n/m) has a well-defined classical limit, and the study addresses the issue of quantum-classical correspondence. Both the perturbative and the nonperturbative regimes are explored. The limitations of the random matrix theory approach are demonstrated.

Nonlinear response of a Kondo system: Perturbation approach to the time-dependent Anderson impurity model

Nonlinear tunneling current through a quantum dot (an Anderson impurity system) subject to both constant and alternating electric fields is studied in the Kondo regime. A systematic diagram technique is developed for perturbation study of the current in physical systems out of equilibrium governed by time - dependent Hamiltonians of the Anderson and the Kondo models. The ensuing calculations prove to be too complicated for the Anderson model, and hence, a mapping on an effective Kondo problem is called for. This is achieved by constructing a time - dependent version of the Schrieffer - Wolff transformation. Perturbation expansion of the current is then carried out up to third order in the Kondo coupling J yielding a set of remarkably simple analytical expressions for the current. The zero - bias anomaly of the direct current differential conductance is shown to be suppressed by the alternating field while side peaks develop at finite source - drain voltage. Both the direct component and the first harmonics of the time - dependent response are equally enhanced due to the Kondo effect, while amplitudes of higher harmonics are shown to be relatively small. A zero alternating bias anomaly is found in the alternating current differential conductance, that is, it peaks around zero alternating bias. This peak is suppressed by the constant bias. No side peaks show up in the differential alternating - conductance but their counterpart is found in the derivative of the alternating current with respect to the direct bias. The results pertaining to nonlinear response are shown to be valid also below the Kondo temperature.

Second- and third-order spin-orbit contributions to nuclear shielding tensors

We present analytical calculations of the electronic spin–orbit interaction contribution to nuclear magnetic shielding tensors using linear and quadratic response theory. The effects of the Fermi contact and the spin-dipole interactions with both the one- and two-electron spin–orbit Hamiltonians, included as first-order perturbations, are studied for the H2X (X=O, S, Se, and Te), HX (X=F, Cl, Br, and I), and CH3X (X=F, Cl, Br, and I) systems using nonrelativistic multiconfiguration self-consistent field reference states. We also present the first correlated study of the spin–orbit-induced contributions to shielding tensors arising from the magnetic field dependence of the spin–orbit Hamiltonian. While the terms usually considered are formally calculated using third-order perturbation theory, the magnetic-field dependent spin-orbit Hamiltonian requires a second-order calculation only. For the hydrogen chalcogenides, we show that contributions often neglected in studies of spin–orbit effects on nuclear shieldings, the spin-dipole coupling mechanism and the coupling of the two-electron spin–orbit Hamiltonian to the Fermi-contact operator, are important for the spin–orbit effect on the heavy-atom shielding, adding up to about half the value of the one-electron spin–orbit interaction with the Fermi-contact contribution. Whereas the second-order spin-orbit-induced shieldings of light ligands are small, the effect is larger for the heavy nuclei themselves and of opposite sign compared to the third-order contribution.

A derivation of centroid molecular dynamics and other approximate time evolution methods for path integral centroid variables

Several methods to approximately evolve path integral centroid variables in real time are presented in this paper, the first of which, the centroid molecular dynamics (CMD) method, is recast into the new formalism of the preceding paper and thereby derived. The approximations involved in the CMD method are thus fully characterized by mathematical derivations. Additional new approaches are also presented: centroid Hamiltonian dynamics (CHD), linearized quantum dynamics (LQD), and a perturbative correction of the LQD method (PT-LQD). The CHD method is shown to be a variation of the CMD method which conserves the approximate time dependent centroid Hamiltonian. The LQD method amounts to a linear approximation for the quantum Liouville equation, while the PT-LQD method includes a perturbative correction to the LQD method. All of these approaches are then tested for the equilibrium position correlation functions of three different one-dimensional nondissipative model systems, and it is shown that certain quantum effects are accounted for by all of them, while others, such as the long time coherence characteristic of low-dimensional nondissipative systems, are not. The CMD method is found to be consistently better than the LQD method, while the PT-LQD method improves the latter and is better than the CMD method in most cases. The CHD method gives results complementary to those of the CMD method.

Experimental Born–Oppenheimer Potential for theX1Σ+Ground State of HeH+: Comparison with theAb InitioPotential

All literature on vibration–rotational and pure rotational transition energies for theX1Σ+state of4HeH+,3HeH+,4HeD+, and3HeD+have been employed in a direct least-squares fit of the radially dependent Hamiltonian. The Born–Oppenheimer potential is represented by a modified Lennard–Jones function that provides for correct behavior in the near dissociation long-range region. Only 19 adjustable parameters were required to represent the 166 measured transition energies, which are reproduced almost to within the measurement accuracies. The fitted potential and Born–Oppenheimer breakdown functions are shown to be in good agreement with the results ofab initiocalculations. The results are used to obtain high-order centrifugal distortion constants as well as level widths for quasi-bound levels.

On the quantum field operator dynamics: memory dependent projected representation

The projection formalism of Zwanzig and Feshbach is used to construct the Heisenberg picture of time dependent field operators in two mutually orthogonal subspaces. A particular form of Zwanzig's generalized master equation involving time dependent Hamiltonians and Liouville superoperators is obtained, from which it is verified that the emerging memory superoperator must satisfy the Volterra integral equation whose kernels involve superoperators which couple the two subspaces at a common instant of time. The subsequent development to describe the time dependence involves time-order cosine and sine projected memory superoperators. It is also shown that this projection separation admits a coupling–decoupling structure for the total (sum over the two subspaces) field operator time evolution which reveals the flux of density from one subspace to the other. The present development to obtain evolution equations for field operators allows a partition of the p-particle reduced density matrix master equation into relevant and irrelevant parts. Such a separation is exact and represents the time dependent fluctuations around the static density coming from the interference terms between the Hamiltonian states in the Liouville space.

Adiabatic invariants and scalar fields in a de Sitter space-time

The method of adiabatic invariants for time dependent Hamiltonians is applied to a massive scalar field in a de Sitter space-time. The scalar field ground state, its Fock space and coherent states are constructed and related to the particle states. Diverse quantities of physical interest are illustrated and compared with previous results.

Convergence properties of the cluster expansion for equal- time Green functions in scalar theories

We investigate the convergence properties of the cluster expansion of equal-time Green functions in scalar theories with quartic self-coupling in (0 + 1), (1 + 1), and (2 + 1) space-time dimensions. The computations are carried out within the equal-time correlation dynamics approach, which consists in a closed set of coupled equations of motion for connected Green functions as obtained by a truncation of the BBGKY hierarchy. We find that the cluster expansion shows good convergence as long as the system is in a localized state (single phase configuration) and that it breaks down in a non-localized state (two phase configuration), as one would naively expect. Furthermore, in the case of dynamical calculations with a time dependent Hamiltonian for the evaluation of the effective potential we find two timescales determining the adiabaticity of the propagation; these are the time required for adiabaticity in the single phase region and the time required for tunneling into the non-localized lowest energy state in the two phase region. Our calculations show a good convergence for the effective potentials in (1 + 1) and (2+1) space-time dimensions since tunneling is suppressed in higher space-time dimensions.