Unitary Time Evolution Operator Research Articles

Nonclassical Effects of a Two-Level Spin System Interacting with a Two-Mode Cavity Field via Two-Photon Transition

The interaction of a two-level cyclic XY n-spin model with a two-mode cavity field involving two-photon transitions is investigated through a generalized Jaynes-Cummings model in the rotating-wave approximation. The two-photon interacting Hamiltonian becomes from the replacement of each single-mode field in the one-photon interacting Hamiltonian with the second-harmonic generation. It was assumed that initially the correlated field modes are in disentangled coherent states having the same photon distribution and that the spin system is in an excited state. At any time t > 0, the spin system and the field are in an entangled state, in this case, via a unitary time evolution operator. Thus, the spontaneous decay of a spin level was treated by considering the interaction of the two-level spin system with the modes of the universe in the vacuum state. The different cases of interest, characterized in terms of a detuning parameter for each mode, which emerge from nonvanishing commutation relations, were analytically implemented and numerically discussed for various values of the initial mean photon number and spin-photon coupling constants. Photon distribution, time evolution of the spin population inversion, as well as the statistical properties of the field leading to the possible production of nonclassical states, such as antibunched light, violations of the Cauchy-Schwartz inequality, and second- and fourth-order squeezing, are examined. The case of zero detuning of both modes was treated in terms of a linearization of the expansion of the time evolution operator, while in other three cases, the computations were conducted via second- and third-order Dyson perturbation expansion of the time evolution operator matrix elements for the excited and ground states of the spin system, respectively.

Dynamics of Two-Level Trapped Ion in a Standing WaveLaser in Noncommutative Space

We study the dynamics of a two-level trapped ion in a standingwave electromagnetic field in two-dimensional (2D) noncommutativespaces in the Lamb–Dicke regime under the rotating waveapproximation. We obtain the explicit analytical expressions forthe energy spectra, energy eigenstates, unitary time evolutionoperator, atomic inversion, and phonon number operators. The Rabioscillations, the collapse, and revivals in the average atomicinversion and the average phonon number are explicitly shown tocontain the information of the parameter of the spacenoncommutativity, which sheds light on proposing new schemes basedon the dynamics of trapped ion to test the noncommutativity.

Pure point spectrum for the time evolution of a periodically rank-N kicked Hamiltonian

We find the conditions under which the spectrum of the unitary time evolution operator for a periodically rank-N kicked system remains pure point. This stability result allows one to analyze the onset of, or lack of chaos in this class of quantum mechanical systems, extending the results for rank-1 systems produced by Combescure and others. This work includes a number of unitary theorems equivalent to those well known and used in the self-adjoint theory.

Weak coherent state path integrals

Weak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise, e.g., in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present article studies the path integral representation of the affine weak coherent state matrix elements of the unitary time-evolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead a well-defined path integral with Wiener measure, based on a continuous-time regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.

Wave-packet dynamics at the mobility edge in two- and three-dimensional systems

We study the time evolution of wave packets at the mobility edge of disordered noninteracting electrons in two and three spatial dimensions. The results of numerical calculations are found to agree with the predictions of scaling theory. In particular, we find that the return probability $P(r=0,t)$ scales like ${t}^{\ensuremath{-}{D}_{2}/d}$, with the generalized dimension of the participation ratio ${D}_{2}$. For long times and short distances the probability density of the wave packet shows power-law scaling $P(r,t)\ensuremath{\propto}{t}^{\ensuremath{-}{D}_{2}/d}{r}^{{D}_{2}\ensuremath{-}d}$. The numerical calculations were performed on network models defined by a unitary time evolution operator providing an efficient model for the study of the wave-packet dynamics.

Network model for a two-dimensional disordered electron system with spin-orbit scattering

We introduce a network model to describe two-dimensional disordered electron systems with spin-orbit scattering. The network model is defined by a discrete unitary time evolution operator. We establish by numerical transfer matrix calculations that the model exhibits a localization-delocalization transition. We determine the corresponding phase diagram in the parameter space of disorder scattering strength and spin-orbit scattering strength. Near the critical point we determine by statistical analysis a one-parameter scaling function and the critical exponent of the localization length to be $\nu=2.51\pm 0.18$. Based on a conformal mapping we also calculate the scaling exponent of the typical local density of states $\alpha_0=2.174 \pm 0.003$.

Time-evolution operator for a forced parametric oscillator

We apply an algebraic technique to describe the evolution of a parametric harmonic oscillator forced by a constant quartic potential. As the first step, we make use of iterative Bogolubov transformations (IBT) to incorporate information from the anharmonic part of the interaction in a nonperturbative form, yielding a unitary time-evolution operator. Later on, we make use of first-order perturbation theory to deal with that part of the interaction which was not incorporated previously. We show numerically that the resulting time-evolution operator is closer to unitarity than is the one obtained if no IBT is applied. The quantum fluctuations of position and momentum are evaluated for “the ground” state. Squeezing and correlation effects are observed. © 1997 John Wiley & Sons, Inc.

Necessary and Sufficient Conditions for Quantum Computation

Abstract Necessary and sufficient conditions are given for a quantummechanical system to possess a coordinate system with respect to which its behaviour at discrete times is that of a universal digital computer. The form of the diagonal representation for the unitary time evolution operator for quantum universal computers is derived; aspects of the transformation between the diagonal representation and the computational representation are shown to be uncomputable. A quantum-mechanical treatment of macroscopic, dissipative computers is given.

LOCALIZATION AT WEAK DISORDER: SOME ELEMENTARY BOUNDS

An elementary proof is given of localization for linear operators H = Ho + λV, with Ho translation invariant, or periodic, and V (·) a random potential, in energy regimes which for weak disorder (λ → 0) are close to the unperturbed spectrum σ (Ho). The analysis is within the approach introduced in the recent study of localization at high disorder by Aizenman and Molchanov [4]; the localization regimes discussed in the two works being supplementary. Included also are some general auxiliary results enhancing the method, which now yields uniform exponential decay for the matrix elements <0|P[a,b] exp (−itH)|x> of the spectrally filtered unitary time evolution operators, with [a, b] in the relevant range.

Perturbation Dynamics of the Infinite Dicke Model

Abstract By means of operator algebraic methods the dynamics of the Dicke model is investigated in the limit where the number of the two-level atoms goes to infinity and the interaction strength remains on a finite level. The infinite atomic system is treated as a mean field quantum lattice system. It is shown that the limiting dynamics is essentially determined by the collective behaviour of the atoms. With Trotter's product formula and perturbation theoretical methods we obtain explicit expressions for the unitary time evolution operators in the uncoupled representation.

The functional integral for quantum systems with Hamiltonians unbounded from below

A quantum system with a self-adjoint Hamiltonian has a well-defined unitary time evolution operator even if there is no ground state. However, if the spectrum of the Hamiltonian is unbounded from below, the Brownian (or imaginary time) functional integral may diverge for all times. Working in one-dimensional non-relativistic quantum mechanics with potentials which are unbounded from below at infinity and at the origin, we show how to regulate the Brownian functional integral, analytically continue to quantum mechanical time, and then remove the cutoff to obtain the quantum Green's function associated with any self-adjoint Hamiltonian. Our constructions illustrate the equivalence of the operator and functional integral approaches to quantum mechanics.

Some remarks on the unitary time evolution operator of the quantum-mechanical forced oscillator. The Weierstrass propagator

The convergence properties of the unitary time evolution operator of the quantum-mechanical forced oscillator are investigated in some detail. On using the inversion of the Weierstrass transform, the propagator of this system is obtained, from which a complete set of square integrable solutions is generated.

Pedagogical aspects of time-dependent rotation operators

For tutorial purposes, the simple problem of a classical magnetic moment interacting with various magnetic field configuration is reformulated in terms of coordinate-free, time-dependent rotation operators. In this approach, the corresponding equation of motion assumes the form of an operator equation, identical with the Schrödinger equation of a spin-1 interacting with the same field configurations. It is solved utilizing deliberately the concepts of a Hermitian Hamiltonian and a unitary time evolution operator to familiarize students early in their curriculum with these important ingredients of the quantum-mechanical formalism. It turns out that besides furnishing useful exercises for the manipulation of three-dimensional rotation operators, being an end in itself, the approach also provides simple but nontrivial examples for a number of theorems for which the students will have frequent use in their quantum-mechanics course.

A finite quantum electrodynamical base for many photon collective phenomena

We analyze the interaction of many ’’nonoverlapping’’ atoms with the common radiation field of transverse photons over a continuum of modes. Typical effects of quantum optics arising from this interaction require extensions to very high powers in the coupling constant e and diverge therefore within a systematic perturbation approach. Since renormalization to such high orders is not feasible in practice, we formulate for the purpose of quantum optics ’’a new peace treaty between QED and its infinities.’’ This is a compromise between the necessity of including effects of very high powers in e and the trivial demands for finite results. We show that a unitary time evolution operator exists if, in essence, only any finite number of levels of each atom is treated as ’’existent.’’ We convince ourselves that the limits of the concept of ’’nonoverlapping atoms’’ are reached long before any infinities can enter. The resulting theories meet still reasonable requirements with respect to the dualism of light, Dicke principle, and causality. They are formulated so that some practical demands of quantum optics (homogeneous and inhomogeneous line broadening, Doppler effects) can be met easily if desired.

On the existence of Weisskopf-Wigner type theories

It is shown that certain features of the Weisskopf-Wigner theory of natural line width can be generalized for the construction of a systematic, mathematically and logically autonomous approximation scheme for treating the interaction of a bound electron with photons. This scheme is equipped with its own hierarchy of orders which are not ruled by powers of the coupling constant. Under very weak conditions on the bound electron all Weisskopf-Wigner theories of finite order exist as ordinary quantum theories on ordinary Hilbert spaces HI and have strictly unitary time evolution operators UI(t), t< infinity . This implies in particular that in such theories the interaction of any finite number of states of the Dirac hydrogen atom with any bounded number of photons can never lead to divergencies. If an infrared catastrophe is banned by a small photon mass mu >0, Weisskopf-Wigner type theories of the interaction of an 'm-level atom', m< infinity , with any unbounded number of photons also exist and have strictly unitary time evolution operators. Some examples of novel applications of Wiesskopf-Wigner type theories are given.

Exact dynamics of an infinite-atom Dicke maser model

We study the dynamics of the Dicke maser model in the limit as the number of atoms becomes infinite and the coupling constant between the atoms and the radiation field goes to zero. We find that the limiting Hamiltonian is integrable and obtain an explicit closed form for the unitary time evolution operators. As a corollary we show that in the limit the radiation emitted by the model is coherent in the sense by Glauber.

Quantum theory of particles with spin zero and one half in external fields

The unitary (pseudo unitary) time-evolution operator for a particle with spin half (zero) in an external time-dependent electromagnetic (scalar) field is used to generate a Bogoliubov automorphism on the algebra of the free in field. For the case of an electric external field (scalar field) a finite expression for Ωout is given and theS-matrix constructed. The latter is unitary and implements the Bogoliubov automorphism. Theorems by Shale and Stinespring are rederived.