Quantum tunneling as a classical anomaly

Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: Complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im(E) -> 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as ReE increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.Comment: 12 pages, 7 figure

Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.Comment: 22 page, 16 figure

Classical Trajectories for Complex Hamiltonians

It has been found that complex non-Hermitian quantum-mechanical Hamiltonians may have entirely real spectra and generate unitary time evolution if they possess an unbroken \cP\cT symmetry. A well-studied class of such Hamiltonians is $H= p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$). This paper examines the underlying classical theory. Specifically, it explores the possible trajectories of a classical particle that is governed by this class of Hamiltonians. These trajectories exhibit an extraordinarily rich and elaborate structure that depends sensitively on the value of the parameter $\epsilon$ and on the initial conditions. A system for classifying complex orbits is presented.Comment: 24 pages, 34 figure

Effects of temperature and carbon source on the isotopic fractionations associated with O_2 respiration for ^(17)O/^(16)O and ^(18)O/^(16)O ratios in E. coli

^(18)O/^(16)O and ^(17)O/^(16)O ratios of atmospheric and dissolved oceanic O_2 are used as biogeochemical tracers of photosynthesis and respiration. Critical to this approach is a quantitative understanding of the isotopic fractionations associated with production, consumption, and transport of O_2 in the ocean both at the surface and at depth. We made measurements of isotopic fractionations associated with O_2 respiration by E. coli. Our study included wild-type strains and mutants with only a single respiratory O_2 reductase in their electron transport chains (either a heme-copper oxygen reductase or a bd oxygen reductase). We tested two common assumptions made in interpretations of O_2 isotope variations and in isotope-enabled models of the O_2 cycle: (i) laboratory-measured respiratory ^(18)O/^(16)O isotopic fractionation factors (^(18)α) of microorganisms are independent of environmental and experimental conditions including temperature, carbon source, and growth rate; And (ii) the respiratory ‘mass law’ exponent, θ, between ^(18)O/^(16)O and ^(17)O/^(16)O, ^(17)α = (^(18)α)^θ, is universal for aerobic respiration. Results demonstrated that experimental temperatures have an effect on both ^(18)α and θ for aerobic respiration. Specifically, lowering temperatures from 37 to 15 °C decreased the absolute magnitude of ^(18)α by 0.0025 (2.5‰), and caused the mass law slope to decrease by 0.005. We propose a possible biochemical basis for these variations using a model of O_2 reduction that incorporates two isotopically discriminating steps: the reversible binding and unbinding of O_2 to a terminal reductase, and the irreversible reduction of that O_2 to water. Finally, we cast our results in a one-dimensional isopycnal reaction-advection-diffusion model, which demonstrates that enigmatic δ^(18)O and Δ^(17)O variations of dissolved O_2 in the dark ocean can be understood by invoking the observed temperature dependence of these isotope effects

Quantum effects in classical systems having complex energy

On the basis of extensive numerical studies it is argued that there are strong analogies between the probabilistic behavior of quantum systems defined by Hermitian Hamiltonians and the deterministic behavior of classical mechanical systems extended into the complex domain. Three models are examined: the quartic double-well potential $V(x)=x^4-5x^2$, the cubic potential $V(x)=frac{1}{2}x^2-gx^3$, and the periodic potential $V(x)=-\cos x$. For the quartic potential a wave packet that is initially localized in one side of the double-well can tunnel to the other side. Complex solutions to the classical equations of motion exhibit a remarkably analogous behavior. Furthermore, classical solutions come in two varieties, which resemble the even-parity and odd-parity quantum-mechanical bound states. For the cubic potential, a quantum wave packet that is initially in the quadratic portion of the potential near the origin will tunnel through the barrier and give rise to a probability current that flows out to infinity. The complex solutions to the corresponding classical equations of motion exhibit strongly analogous behavior. For the periodic potential a quantum particle whose energy lies between -1 and 1 can tunnel repeatedly between adjacent classically allowed regions and thus execute a localized random walk as it hops from region to region. Furthermore, if the energy of the quantum particle lies in a conduction band, then the particle delocalizes and drifts freely through the periodic potential. A classical particle having complex energy executes a qualitatively analogous local random walk, and there exists a narrow energy band for which the classical particle becomes delocalized and moves freely through the potential.Comment: 16 pages, 12 figure

Bounding biomass in the Fisher equation

The FKPP equation with a variable growth rate and advection by an incompressible velocity field is considered as a model for plankton dispersed by ocean currents. If the average growth rate is negative then the model has a survival-extinction transition; the location of this transition in the parameter space is constrained using variational arguments and delimited by simulations. The statistical steady state reached when the system is in the survival region of parameter space is characterized by integral constraints and upper and lower bounds on the biomass and productivity that follow from variational arguments and direct inequalities. In the limit of zero-decorrelation time the velocity field is shown to act as Fickian diffusion with an eddy diffusivity much larger than the molecular diffusivity and this allows a one-dimensional model to predict the biomass, productivity and extinction transitions. All results are illustrated with a simple growth and stirring model.Comment: 32 Pages, 13 Figure

Complexified dynamical systems.

Accepted versio

Modified spectrum autointerferometric correlation (MOSAIC) for single-shot pulse characterization

A method for generation of the modified spectrum autointerferometric correlation that allows single-shot pulse characterization is demonstrated. A sensitive graphical representation of the ultrashort pulse phase quality is introduced that delineates the difference between the presence of temporal and spectral phase distortions. Using these schemes, full-field reconstruction of ultrashort laser pulses is obtained in real time using an efficient iterative technique. (a) Single-shot characterization using a combination of fringe-free (noninterferometric) autocorrelation and second-harmonic spectrum (b) A hybrid graphical representation that distinguishes between spectral and temporal phase distortions (c) Real-time full-field reconstruction using the above schemes with an efficient sequential search algorithm Naganuma et al. showed that the pulse spectrum and IAC provide a sufficient dataset to uniquely reconstruct the complex electric field, with only a timedirection ambiguity The increased SNR found on averaged MOSAIC traces extends the utility of all retrieval techniques using the dataset outlined by Naganuma et al. The principle of computing a MOSAIC can be described in the frequency domain as follows: a secondorder IAC waveform with a fringe frequency ⍀ is Fourier transformed to generate a spectrum. Spectral filtering is then performed to remove the ⍀ component and amplify the 2⍀ component by a factor of 2. An inverse Fourier transform generates a new time-domain signal known as a fringe-resolved MOSAIC In the (delay) time-domain analysis, the maximum and minimum envelopes of MOSAIC are given by the intensity autocorrelation, g͑͒ = ͐f͑t͒f͑t + ͒dt, and the difference computation, S min = g͑͒ − ͉g p ͉͑͒, respectivel

Exact Isospectral Pairs of PT-Symmetric Hamiltonians

A technique for constructing an infinite tower of pairs of PT-symmetric Hamiltonians, $\hat{H}_n$ and $\hat{K}_n$ (n=2,3,4,...), that have exactly the same eigenvalues is described. The eigenvalue problem for the first Hamiltonian $\hat{H}_n$ of the pair must be posed in the complex domain, so its eigenfunctions satisfy a complex differential equation and fulfill homogeneous boundary conditions in Stokes' wedges in the complex plane. The eigenfunctions of the second Hamiltonian $\hat{K}_n$ of the pair obey a real differential equation and satisfy boundary conditions on the real axis. This equivalence constitutes a proof that the eigenvalues of both Hamiltonians are real. Although the eigenvalue differential equation associated with $\hat{K}_n$ is real, the Hamiltonian $\hat{K}_n$ exhibits quantum anomalies (terms proportional to powers of $\hbar$). These anomalies are remnants of the complex nature of the equivalent Hamiltonian $\hat{H}_n$. In the classical limit in which the anomaly terms in $\hat{K}_n$ are discarded, the pair of Hamiltonians $H_{n,classical}$ and $K_{n,classical}$ have closed classical orbits whose periods are identical.Comment: 18 pages, 12 figure

Complex trajectories of a simple pendulum

Accepted versio