Unitary Propagator Research Articles

A random matrix approach to long range acoustic propagation in the ocean.

Low frequency sound propagates in the ocean within a wave guide, formed by the confining effects of temperature, salinity, and pressure on the sound speed. Within the wave guide, sound scatters due to range dependent sound speed oscillations from internal waves. These weak perturbations in the sound speed serve to randomize the acoustic signal so that the structure formed by the time series of arrivals at the receivers (i.e., the timefront) contains only minimal average information about the propagation through the action of a generalized central limit theorem. This study characterizes this information by the parameters of a statistical ensemble model for the propagation. The propagation is described by the evolution of a unitary propagation matrix, whose elements are the complex probability amplitudes for modal transitions during the propagation. The ensemble model is constructed from a product of unitary random matrices utilizing complex Gaussian random variables with minimal information about the propagation to 50 km stored in a matrix of variances and a vector of mean phases. A comparison of the properties of the average intensity timefront resulting from this ensemble model is made with those from the simulated propagations for several ranges. [This work was supported by ONR and NSF.]

The Liouville equation for singular ergodic magnetic Schrödinger operators

We study the time evolution of a density matrix in a quantum mechanical system described by an ergodic magnetic Schrödinger operator with singular magnetic and electric potentials, the electric field being introduced adiabatically. We construct a unitary propagator that solves weakly the corresponding time-dependent Schrödinger equation and solve a Liouville equation in an appropriate Hilbert space.

Time-Dependent Density Functional Theory for Open Quantum Systems with Unitary Propagation

We extend the Runge-Gross theorem for a very general class of open quantum systems under weak assumptions about the nature of the bath and its coupling to the system. We show that for Kohn-Sham (KS) time-dependent density functional theory, it is possible to rigorously include the effects of the environment within a bath functional in the KS potential. A Markovian bath functional inspired by the theory of nonlinear Schrödinger equations is suggested, which can be readily implemented in currently existing real-time codes. Finally, calculations on a helium model system are presented.

Topology of the quantum control landscape for observables

A broad class of quantum control problems entails optimizing the expectation value of an observable operator through tailored unitary propagation of the system density matrix. Such optimization processes can be viewed as a directed search over a quantum control landscape. The attainment of the global extrema of this landscape is the goal of quantum control. Local optima will generally exist, and their enumeration is shown to scale factorially with the system's effective Hilbert space dimension. A Hessian analysis reveals that these local optima have saddlepoint topology and cannot behave as suboptimal extrema traps. The implications of the landscape topology for practical quantum control efforts are discussed, including in the context of nonideal operating conditions.

Single-bit feedback and quantum-dynamical decoupling

Synthesizing an effective identity evolution in a target system subjected to unwanted unitary or non-unitary dynamics is a fundamental task for both quantum control and quantum information processing applications. Here, we investigate how single-bit, discrete-time feedback capabilities may be exploited to enact or to enhance quantum procedures for effectively suppressing unwanted dynamics in a finite-dimensional open quantum system. An explicit characterization of the joint unitary propagators correctable by a single-bit feedback strategy for arbitrary evolution time is obtained. For a two-dimensional target system, we show how by appropriately combining quantum feedback with dynamical decoupling methods, concatenated feedback-decoupling schemes may be built, which can operate under relaxed control assumptions and can outperform purely closed-loop and open-loop protocols.

A lower bound for the Sturm–Liouville eigenvalue problem on a quantum computer

We study the complexity of approximating the smallest eigenvalue of a univariate Sturm–Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schrödinger equation with a given potential for the ground state energy. The Sturm–Liouville problem depends on a function q, which, in the case of the Schrödinger equation, can be identified with the potential function V . Recently Papageorgiou and Woźniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical worst-case and randomized settings for smooth functions q. Their method uses the (discretized) unitary propagator and arbitrary powers of it as a query (“power queries”). They showed that the Sturm–Liouville equation can be solved with O ( log ( 1 / ε ) ) power queries, while the number of queries in the worst-case and randomized settings on a classical computer is polynomial in 1 / ε . This proves that a quantum computer with power queries achieves an exponential reduction in the number of queries compared to a classical computer. In this paper we show that the number of queries in Papageorgiou's and Woźniakowski's algorithm is asymptotically optimal. In particular we prove a matching lower bound of Ω ( log ( 1 / ε ) ) power queries, therefore showing that Θ ( log ( 1 / ε ) ) power queries are sufficient and necessary. Our proof is based on a frequency analysis technique, which examines the probability distribution of the final state of a quantum algorithm and the dependence of its Fourier transform on the input.

A lower bound for the Sturm-Liouville eigenvalue problem on a quantum computer

We study the complexity of approximating the smallest eigenvalue of a univariate Sturm-Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schrodinger equation with a given potential for the ground state energy.The Sturm-Liouville problem depends on a function q, which, in the case of the Schrodinger equation, can be identified with the potential function V. Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical worst-case and randomized settings for smooth functions q. Their method uses the (discretized) unitary propagator and arbitrary powers of it as a query (power queries). They showed that the Sturm-Liouville equation can be solved with O(log(1/e)) power queries, while the number of queries in the worst-case and randomized settings on a classical computer is polynomial in 1/e. This proves that a quantum computer with power queries achieves an exponential reduction in the number of queries compared to a classical computer.In this paper we show that the number of queries in Papageorgiou's and Wozniakowski's algorithm is asymptotically optimal. In particular we prove a matching lower bound of Ω(log(1/e)) power queries, therefore showing that Θ(log(1/e)) power queries are sufficient and necessary. Our proof is based on a frequency analysis technique, which examines the probability distribution of the final state of a quantum algorithm and the dependence of its Fourier transform on the input.

Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms

In this paper, we introduce optimal control algorithm for the design of pulse sequences in NMR spectroscopy. This methodology is used for designing pulse sequences that maximize the coherence transfer between coupled spins in a given specified time, minimize the relaxation effects in a given coherence transfer step or minimize the time required to produce a given unitary propagator, as desired. The application of these pulse engineering methods to design pulse sequences that are robust to experimentally important parameter variations, such as chemical shift dispersion or radiofrequency (rf) variations due to imperfections such as rf inhomogeneity is also explained.

Representing multiqubit unitary evolutions via Stokes tensors

For the tensor of coherences parametrization of a multiqubit density operator, we provide an explicit formulation of the corresponding unitary dynamics at infinitesimal level. The main advantage of this formalism (clearly reminiscent of the idea of ``coherences'' and ``coupling Hamiltonians'' of spin systems) is that the pattern of correlation between qubits and the pattern of infinitesimal correlation are highlighted simultaneously and can be used constructively for qubit manipulation. For example, it allows to compute explicitly a Rodrigues' formula for the one-parameter orbits of nonlocal Hamiltonians. The result is easily generalizable to orbits of Cartan subalgebras and allows to write the Cartan decomposition of unitary propagators as a linear action.

Quantum hydrodynamics: application to N-dimensional reactive scattering.

The quantum hydrodynamic equations associated with the de Broglie-Bohm formulation of quantum mechanics are solved using a new methodology which gives an accurate, unitary, and stable propagation of a time dependent quantum wave packet [B. K. Kendrick, J. Chem. Phys. 119, 5805 (2003)]. The methodology is applied to an N-dimensional model chemical reaction with an activation barrier. A parallel version of the methodology is presented which is designed to run on massively parallel supercomputers. The computational scaling properties of the parallel code are investigated both as a function of the number of processors and the dimension N. A decoupling scheme is introduced which decouples the multidimensional quantum hydrodynamic equations into a set of uncoupled one-dimensional problems. The decoupling scheme dramatically reduces the computation time and is highly parallelizable. Furthermore, the computation time is shown to scale linearly with respect to the dimension N=2,...,100.

From classical mechanics to unitary propagation without quantization

In the 1970s Marc Kac demonstrated that a simple stochastic model involving random walks could provide a microscopic basis for the telegraph equations in the same way that Brownian motion provided a stochastic basis for the diffusion equation. This article demonstrates that a modification of the space-time geometry of the random walks in the Kac model provides the Dirac equation with a similar microscopic basis. The central feature of the new model is that space-time paths occur in ‘entwined pairs’; for each path in the Kac model there is a corresponding ‘return-path’ which is traversed in the opposite direction with respect to macroscopic time. The special pairing of paths automatically ‘quantizes’ the system eliminating the need for formal analytic continuation and providing an explicit mechanism for ‘wave-particle duality’. The result brings the Dirac equation into the domain of classical statistical mechanics where it can be studied as a phenomenology with a known stochastic basis.

Controllability of quantum mechanical systems by root space decomposition of su(N)

The controllability property of the unitary propagator of an N-level quantum mechanical system subject to a single control field is described using the structure theory of semisimple Lie algebras. Sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases.

Stability of quantum motion and correlation decay

We derive a simple and general relation between the fidelity of quantum motion, characterizing the stability of quantum dynamics with respect to arbitrary static perturbation of the unitary evolution propagator, and the integrated time auto-correlation function of the generator of perturbation. Surprisingly, this relation predicts the slower decay of fidelity the faster decay of correlations is. In particular, for non-ergodic and non-mixing dynamics, where asymptotic decay of correlations is absent, a qualitatively different and faster decay of fidelity is predicted on a time scale 1/delta as opposed to mixing dynamics where the fidelity is found to decay exponentially on a time-scale 1/delta^2, where delta is a strength of perturbation. A detailed discussion of a semi-classical regime of small effective values of Planck constant is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behavior is recovered in the classical limit hbar->0, as the two limits delta->0 and hbar->0 do not commute. In addition we also discuss non-trivial dependence on the number of degrees of freedom. All the theoretical results are clearly demonstrated numerically on a celebrated example of a quantized kicked top.

Time optimal control in spin systems

In this paper, we study the design of pulse sequences for NMR spectroscopy as a problem of time optimal control of the unitary propagator. Radio frequency pulses are used in coherent spectroscopy to implement a unitary transfer of state. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. Moreover, the general proofs are no more difficult than the specific problems of current interest. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group, what is the minimum time required to steer the system from some initial point to a specified final point? In NMR spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two spin system.

Hydrogen wavefunction in intense laser fields: A unitary integrator for a high performance parallel computer

The time evolution of the ground state of the hydrogen atom in a very intense, linearly polarized, pulsed laser field is investigated by solving the two-dimensional Schrödinger equation. We use a unitary propagation scheme based on multiple application of the split operator formalism. For a realistic description, the Coulomb singularity is not cut off but treated exactly using fast sine transforms to switch between direct and Fourier space. The implementation and performance enhancement through parallelization on a Cray T3E are discussed.

The effect of stochastic rough interfaces on coupled-mode elastic waves

Summary The effect of stochastic fluctuations of the interface boundaries has been incorporated into the elastic coupled-mode equations for 2-D range-dependent media. We assume the medium to be characterized by some deterministic range-dependent layered structure superposed by stochastic boundary fluctuations. The deterministic range-dependent structure defines the reference structure, and the reference structure with superposed stochastic fluctuations defines the true layer boundaries in the medium. The boundary conditions on the true boundary are expanded about the reference boundary, and first-order perturbation theory is applied to the boundary conditions, as well as to the equation of motion. The calO(1) system and the calO(ɛ) system represent wave propagation in the deterministic model and in the stochastic model, respectively, with a first-order perturbation of roughness. The solution of the calO(1) system is referred to as the primary field. The solutions to the calO(ɛ) system relate the coherent field and the scattered field, which are represented as the superposition of local modes multiplied by the stochastic modal amplitudes. Propagation of coupled-mode elastic waves in a 2-D deterministic range-dependent medium is represented by a unitary coupled-mode propagator. The evolution equation for the stochastic medium and the stochastic coupling matrices, which acts to convert the coherent field to the scattered field, is derived. Enforcing energy conservation on the calO(ɛ) system leads to the requirement that the propagator for the stochastic medium must satisfy a Lippmann–Schwinger-type integral equation, whose solution can be represented by a formal perturbation series for the multiply scattered wavefields. The integral equations for the mean field and the covariance of the field are also presented. These formal theoretical results are valid to all orders of multiple scattering. In the second part of the paper the Born approximation to the Lippmann–Schwinger integral equation is used to extract information about attenuation due to rough surface scattering and in the design of an inverse problem for the boundary roughness variance and correlation length. By defining the modal scattering cross-section, the formula for the scattering Q− 1s from the Born approximation for the scattered field is derived. The modal scattering cross-section and scattering Qs for a range-dependent model with stochastic roughness are computed for both exponential and Gaussian correlation functions. Finally, a formula for the power spectrum of the coherent field is derived from the Born approximation, and an inversion for the roughness variance and correlation length is designed by power spectrum fitting.

A Canonical Quantization of the Baker's Map

We present here a canonical quantization for the baker's map. The method we use is quite different from that used in Balazs and Voros and Saraceno. We first construct a natural “baker covering map” on the plane R2. We then use as the quantum algebra of observables the subalgebra of operators onL2(R) generated by {exp(2πix), exp(2πip)}. We construct a unitary propagator such that as ħ→0 the classical dynamics is returned. For Planck's constanth=1/N, we show that the dynamics can be reduced to the dynamics on anN-dimensional Hilbert space, and the unitaryN×Nmatrix propagator is the same as given by Balazs and Voros, except for a small correction of orderh. This correction is shown to preserve the classical symmetryx→1−xandp→1−pin the quantum dynamics for periodic boundary conditions.

Mixed finite element beam propagation method

An efficient mixed finite element (FE) beam propagation method (BPM) for three-dimensional (3-D) simulations is developed for integrated optic devices. Wide angle propagation is allowed by applying Pade approximants to the finite element operator. Mixed finite elements prevent spurious modes and accurately model waveguide corners allowing the adequate description of polarization effects. Furthermore, the finite element matrices are Hermitian leading to a unitary propagation scheme if lossless waveguides surrounded by metallic walls are assumed. In contrast to finite difference schemes energy conservation holds explicitly.

Classical limits for quantum maps on the torus

We provide a rigorous canonical quantization for the following toral automorphisms: cat maps, generalized kicked maps, and generalized Harper maps. For each of these systems we construct a unitary propagator and prove the existence of a well-defined classical limit. We also provide an alternative derivation of Hannay and Berry results for the cat map propagator on the plane.

Mourre theory for time-periodic systems

Abstract.Studies for A.C. Stark Hamiltonian are closely related to that for the self-adjoint operator on torus. In this paper we use Mourre’s commutator method, which makes great progress for the study of time-independent Hamiltonian. By use of it we show the asymptotic behavior of the unitary propagator as σ → ± ∞.