Initial Hamiltonian Research Articles

Testing quantum adiabaticity with quench echo

Adiabaticity of quantum evolution is important in many settings; one example is adiabatic quantum computation (AQC). Nevertheless, to date, there is no effective method available for testing the adiabaticity of the evolution for the case where the eigenenergies of the driven Hamiltonian are not known. We propose a simple method for checking the adiabaticity of a quantum process for an arbitrary quantum system. We further propose an operational method for finding more efficient protocols that approximate adiabaticity, and suggest a ‘uniformly adiabatic’ quench scheme based on the Kibble–Zurek mechanism for the case where the initial and final Hamiltonians are given. This method should help in implementing AQC and other tasks where preserving the system in the ground state of a time-dependent Hamiltonian is desired.

Finite-Temperature Entanglement Dynamics in an Anisotropic Two-Qubit Heisenberg Spin Chain

This paper investigates the entanglement dynamics of an anisotropic two-qubit Heisenberg spin chain in the presence of decoherence at finite temperature. The time evolution of the concurrence is studied for different initial Werner states. The influences of initial purity, finite temperature, spontaneous decay and Hamiltonian on the entanglement evolution are analyzed in detail. Our calculations show that the finite temperature restricts the evolution of the entanglement all the time when the Hamiltonian improves it and the spontaneous decay to the reservoirs can produce quantum entanglement with the anisotropy of spin-spin interaction. Finally, the steady-state concurrence which may remain non-zero for low temperature is also given.

A study of heuristic guesses for adiabatic quantum computation

Adiabatic quantum computation (AQC) is a universal model for quantum computation which seeks to transform the initial ground state of a quantum system into a final ground state encoding the answer to a computational problem. AQC initial Hamiltonians conventionally have a uniform superposition as ground state. We diverge from this practice by introducing a simple form of heuristics: the ability to start the quantum evolution with a state which is a guess to the solution of the problem. With this goal in mind, we explain the viability of this approach and the needed modifications to the conventional AQC (CAQC) algorithm. By performing a numerical study on hard-to-satisfy 6 and 7 bit random instances of the satisfiability problem (3-SAT), we show how this heuristic approach is possible and we identify that the performance of the particular algorithm proposed is largely determined by the Hamming distance of the chosen initial guess state with respect to the solution. Besides the possibility of introducing educated guesses as initial states, the new strategy allows for the possibility of restarting a failed adiabatic process from the measured excited state as opposed to restarting from the full superposition of states as in CAQC. The outcome of the measurement can be used as a more refined guess state to restart the adiabatic evolution. This concatenated restart process is another heuristic that the CAQC strategy cannot capture.

A WEIL REPRESENTATION OF sp(4) REALIZED BY DIFFERENTIAL OPERATORS IN THE SPACE OF SMOOTH FUNCTIONS ON S2 × S1

In the space of complex-valued smooth functions on S2 × S1, we explicitly realize a Weil representation of the real Lie algebra sp(4) by means of differential generators. This representation is a rare example of highest weight irreducible representation of sp(4) all whose weight spaces are 1-dimensional. We also show how this space splits into the direct sum of irreducible sl(2)-submodules. Selected applications: complete classification of yrast-band energies in even-even nuclei, the dynamical symmetry in some collective models of nuclear structure, the mapping methods for simplifying initial problem Hamiltonians.

Adiabatic quantum computation with a one-dimensional projector Hamiltonian

Adiabatic quantum computation is based on the adiabatic evolution of quantum systems. We analyse a particular class of qauntum adiabatic evolutions where either the initial or final Hamiltonian is a one-dimensional projector Hamiltonian on the corresponding ground state. The minimum energy gap which governs the time required for a successful evolution is shown to be proportional to the overlap of the ground states of the initial and final Hamiltonians. We show that such evolutions exhibit a rapid crossover as the ground state changes abruptly near the transition point where the energy gap is minimum. Furthermore, a faster evolution can be obtained by performing a partial adiabatic evolution within a narrow interval around the transition point. These results generalize and quantify earlier works.

Analytical theory for spacecraft motion about Mercury

In the framework of the elliptic restricted three-body problem we develop an analytical theory for spacecraft motion close to Mercury. Besides the perturbations due to the gravity of the Sun and Mercury and the eccentricity of Mercury's orbit around the Sun, i.e., the elliptic restricted three-body problem, the theory includes the effects of the oblateness and the possible latitudinal asymmetry of Mercury, and is valid for any eccentricity of the spacecraft's orbit. The initial Hamiltonian defines a non-autonomous but periodic dynamical system of two degrees of freedom. The mean motion of the spacecraft and the time are averaged using two successive Lie–Deprit transformations. The resulting Hamiltonian defines a one degree of freedom system and depends upon three essential parameters. When the latitudinal asymmetry coefficient vanishes the flow of this system is entirely analyzed through the discussion of the occurrence of its (relative) equilibria and bifurcations in accordance with the parameters the problem depends upon. Frozen orbits of the initial system together with their stability are obtained related to the relative equilibria. If the latitudinal asymmetry of Mercury is taken into account, the equatorial symmetry of the problem is broken and introduces important changes in the dynamics. A variety of tests show a very good agreement between averaged and non-averaged models, and the reliability of the theory is further checked by performing long-term integrations in ephemeris.

Adiabatic approximation with exponential accuracy for many-body systems and quantum computation

We derive a version of the adiabatic theorem that is especially suited for applications in adiabatic quantum computation, where it is reasonable to assume that the adiabatic interpolation between the initial and final Hamiltonians is controllable. Assuming that the Hamiltonian is analytic in a finite strip around the real-time axis, that some number of its time derivatives vanish at the initial and final times, and that the target adiabatic eigenstate is nondegenerate and separated by a gap from the rest of the spectrum, we show that one can obtain an error between the final adiabatic eigenstate and the actual time-evolved state which is exponentially small in the evolution time, where this time itself scales as the square of the norm of the time derivative of the Hamiltonian divided by the cube of the minimal gap.

Collective properties and combined quantum transitions of two‐dimensional magnetoexcitons

AbstractThe article consists of two parts describing two associated properties of two‐dimensional magnetoexcitons. In the first part, the theory of combined exciton‐cyclotron resonance in the quantum well structures is developed for a strong magnetic field. In the first part, we calculate the absorption band structure for optical quantum transitions with circularly polarized radiation creating a two‐dimensional exciton and simultaneously exciting one of the resident electrons from the lowest to the first excited Landau level. In the second part, we discuss the collective elementary excitations of the system of two‐dimensional magnetoexcitons in the ground state of Bose–Einstein condensation on the arbitrary wave vector $\mathop K\limits^ \to$ ≠ 0 in the Hartree–Fock approximation. The breaking of the gauge symmetry of the initial Hamiltonian was introduced following the idea proposed by Bogoliubov in his theory of quasi‐averages. The energy spectrum of the collective elementary excitations is characterized by the interconnection of the exciton and plasmon branches. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

Adiabatic approximation, Gell-Mann and Low theorem, and degeneracies: A pedagogical example

We study a simple system described by a 2x2 Hamiltonian and the evolution of the quantum states under the influence of a perturbation. More precisely, when the initial Hamiltonian is not degenerate,we check analytically the validity of the adiabatic approximation and verify that, even if the evolution operator has no limit for adiabatic switchings, the Gell-Mann and Low formula allows to follow the evolution of eigenstates. In the degenerate case, for generic initial eigenstates, the adiabatic approximation (obtained by two different limiting procedures) is either useless or wrong, and the Gell-Mann and Low formula does not hold. We show how to select initial states in order to avoid such failures.

LIMITATIONS OF SOME SIMPLE ADIABATIC QUANTUM ALGORITHMS

Let H(t) = (1 - t/T)H0 + (t/T)H1, t ∈ [0,T], be the Hamiltonian governing an adiabatic quantum algorithm, where H0 is diagonal in the Hadamard basis and H1 is diagonal in the computational basis. We prove that H0 and H1 must each have at least two large mutually-orthogonal eigenspaces if the algorithm's running time is to be subexponential in the number of qubits. We also reproduce the optimality proof of Farhi and Gutmann's search algorithm in the context of this adiabatic scheme; because we only consider initial Hamiltonians that are diagonal in the Hadamard basis, our result is slightly stronger than the original.

Microcanonical quantum fluctuation theorems

Previously derived expressions for the characteristic function of work performed on a quantum system by a classical external force are generalized to arbitrary initial states of the considered system and to Hamiltonians with degenerate spectra. In the particular case of microcanonical initial states, explicit expressions for the characteristic function and the corresponding probability density of work are formulated. Their classical limit as well as their relations to the corresponding canonical expressions are discussed. A fluctuation theorem is derived that expresses the ratio of probabilities of work for a process and its time reversal to the ratio of densities of states of the microcanonical equilibrium systems with corresponding initial and final Hamiltonians. From this Crooks-type fluctuation theorem a relation between entropies of different systems can be derived which does not involve the time-reversed process. This entropy-from-work theorem provides an experimentally accessible way to measure entropies.

Density of states of two-dimensional aluminum nanoclusters in the Hubbard model

The density of states of a two-dimensional square nanosystem composed of N × N aluminum atoms (N = 3−30) is calculated in the framework of the Hubbard model. It is demonstrated that, at a small parameter N, the density of states depends substantially on the number of atoms and on the position of a particular atom in the lattice. As the parameter N increases, the density of states for the vertex and edge atoms tends to the value of the density of states for the bulk atoms. The temperature of the system is implicitly included by specifying the energy of hopping in the initial Hamiltonian.

Ground states of Heisenberg spin chains via quantum annealing

We suggest using the method of quantum annealing for computing the ground state of the Heisenberg spin chains. Our initial Hamiltonian describes a spin system in a highly non-uniform magnetic field. The initial Hamiltonian gradually transforms into the Heisenberg Hamiltonian, which describes the exchange interaction and the Zeeman interaction with the uniform magnetic field. We demonstrate quantum annealing for 2-, 3-, and 9-spin systems. In particular, we consider the alternate (ferro- and antiferromagnetic) exchange integral. By changing the sign of the exchange integrals we switch from the frustrated to the corresponding non-frustrated spin system. We discuss the similarity and difference between the frustrated and non-frustrated spin systems.

Local field effects and stimulated multimode scattering of resonance radiation in a two-level medium

Scattering of resonance radiation in a dense two-level medium is considered with allowance for the local field effects. The system is studied in the framework of the mean-field approximation for a modified hierarchy of the Bogoliubov-Born-Green-Kirkwood-Yvon equations for reduced density matrices of an ensemble of atoms and modes of a quantized electromagnetic field. The local field correction is consistently derived from the initial Hamiltonian of the system. The problem of scattering of a short rectangular pulse from a quasi-point sample is considered numerically and theoretically. The possibility of a multimode spectrum of scattered radiation with frequencies multiple of the Rabi frequency and with other intermediate frequencies is demonstrated. The relative intensities of spectral lines are determined.

The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems

We consider families of one and a half degrees of freedom rapidly forced Hamiltonian systems which are perturbations of one degree of freedom Hamiltonians with a homoclinic connection. We derive the inner equation for this class of Hamiltonian system which is expressed as the Hamiltonian–Jacobi equation of a one a half degrees of freedom Hamiltonian. The inner equation depends on a parameter not necessarily small.We prove the existence of special solutions of the inner equation with a given behaviour at infinity. We also compute the asymptotic expression for the difference between these solutions. In some perturbative cases, this asymptotic expression is strongly related with the Melnikov function associated with our initial Hamiltonian.

Semiclassical description of a sixth-order quadrupole boson Hamiltonian

A sixth-order quadrupole boson Hamiltonian is treated through a time dependent variational principle approach choosing as trial function a coherent state with respect to zeroth b 0 † and second b 2 † + b −2 † components of the quadrupole bosons. The coefficients involved in the model Hamiltonian are chosen so that the classical effective potential energy term has two distinct minima. The classical system is described by two degrees of freedom and has two constants of motion. The equation of motion for the radial coordinate is analytically solved and the resulting trajectories are extensively studied. One distinguishes three energy regions exhibiting different types of trajectories. When one passes from the region characterized by two wells to the region of energies higher than the maximum value of the effective potential the trajectories period exhibits a singularity which reflects in fact a phase transition. The classical trajectories are quantized by a constraint of the integral action similar to the well known Bohr–Sommerfeld quantization condition. The semiclassical spectra corresponding to the two potential wells have specific properties. The tunneling process through the potential barrier is also studied. It is a remarkable fact that the transmission coefficients exhibit jumps in magnitude when the intrinsic momentum acquires certain values. It is an open question whether such discontinuities in the transmission coefficients show up also when the final states are not bound. Semiclassical energy levels are compared also with the exact eigenvalues of the initial Hamiltonian, obtained through a diagonalization procedure. The present approach suggests that the energy levels in the laboratory frame can be written as a sum of energy associated to the intrinsic degrees of freedom and a rotational term obtainable through a projection method. This is numerically confirmed for the ground band.

Perturbative approach to the adiabatic approximation

A new and intuitive perturbative approach to time-dependent quantum mechanics problems is presented, which is useful in situations where the evolution of the Hamiltonian is slow. The state of a system which starts in an instantaneous eigenstate of the initial Hamiltonian is written as a power series which has a straightforward diagrammatic representation. Each term of the series corresponds to a sequence of ``adiabatic'' evolutions, during which the system remains in an instantaneous eigenstate of the Hamiltonian, punctuated by transitions from one state to another. The first term of this series is the standard adiabatic evolution, the next is the well known first correction to it, and subsequent terms can be written down essentially by inspection. Although the final result is perhaps not terribly surprising, it seems to not be widely known, and the interpretation is new, as far as we know. Application of the method to the adiabatic approximation is given, and some discussion of the validity of this approximation is presented.

Exponential complexity of an adiabatic algorithm for an NP-complete problem

We prove an analytical expression for the size of the gap between the ground and the first excited state of quantum adiabatic algorithm for the 3-satisfiability, where the initial Hamiltonian is a projector on the subspace complementary to the ground state. For large problem sizes the gap decreases exponentially and as a consequence the required running time is also exponential.

Approximating the Invariant Sets of a Finite Straight Segment near Its Collinear Equilibria

This paper reviews a method for computing explicitly the asymptotic expressions of some invariant manifolds in the vicinity of equilibrium points corresponding to Hamiltonian systems of m degrees of freedom. The method is based on the use of generalized normal forms. Our goal is to show the power of this technique by applying it to the study of one of the colinear equilibrium points corresponding to the Hamiltonian system defined by the motion of a particle orbiting around a finite straight segment. We make use of three different Lie transformations: (i) we calculate the Hamilton function corresponding to the center manifold of the colinear equilibrium; (ii) we determine the Hamiltonian related to the stable-unstable direction; and (iii) we obtain the Hamiltonian related to one of the two stable-stable directions. By means of (i), we are able to parameterize the center, the stable, and the unstable manifolds of the original system using the direct changes of coordinates of the two transformations. We also compute the normally hyperbolic invariant manifold associated with the equilibrium, together with its stable and unstable manifolds. Using (ii), we compute the invariant 2-tori and quasi-periodic orbits in the neighborhood of the equilibrium chosen. Through the normal form Hamiltonian obtained by (iii) we are able to determine some periodic orbits of the initial Hamiltonian.

Darboux transformation of the Green’s function of a regular Sturm-Liouville problem

Darboux transformations of a regular Sturm-Liouville problem are considered. A relationship between the Green’s functions of the initial and transformed problems is found. Second-order reducible and irreducible Darboux transformations are analyzed. It is demonstrated that a new second-order irreducible supersymmetry can exist in the regular case for which the spectrum of the intermediate Hamiltonian differs radically from the spectrum of the initial Hamiltonian. The results obtained are illustrated by the example of a particle in a box.