Simulations of continuum coherent states and its use in quantum cryptographic systems

Abstract

The continuum states formalism is suitable for field quantization in optical fibre; however, they are harder to use than discrete states. On the other hand, a Hermitian phase operator can be defined only in a finite dimensional space. We approximated a coherent continuum state by a finite tensor product of coherent states, each one defined in a finite dimensional space. Using this, in the correct limit, we were able to obtain some statistical properties of the photon number and phase of the continuum coherent states from the probability density functions of the individual, finite dimensional, coherent states. Then, we performed a simulation of the BB84 protocol, using the continuum coherent states, in a fibre interferometer commonly used in quantum cryptography. We observed the fluctuations of the mean photon number in the pulses that arrive at Bob, which occurs in the practical system, introduced by the statistical property of the simulation.     

On the quantumness of correlations in nuclear magnetic resonance

Nuclear magnetic resonance (NMR) was successfully employed to test several protocols and ideas in quantum information science. In most of these implementations, the existence of entanglement was ruled out. This fact introduced concerns and questions about the quantum nature of such bench tests. In this paper, we address some issues related to the non-classical aspects of NMR systems. We discuss some experiments where the quantum aspects of this system are supported by quantum correlations of separable states. Such quantumness, beyond the entanglement-separability paradigm, is revealed via a departure between the quantum and the classical versions of information theory. In this scenario, the concept of quantum discord seems to play an important role. We also present an experimental implementation of an analogue of the single-photon Mach-Zehnder interferometer employing two nuclear spins to encode the interferometric paths. This experiment illustrates how non-classical correlations of separable states may be used to simulate quantum dynamics. The results obtained are completely equivalent to the optical scenario, where entanglement (between two field modes) may be present.     

Improved method for calibrating a Stokes polarimeter

We present a method for calibrating a polarization state analyzer that uses a set of well- characterized reference polarization states and makes no assumptions about the optics contained in the polarimeter other than their linearity. The method requires that a matrix be constructed that contains the data acquired for each of the reference polarization states and that this matrix be pseudoinverted. Since this matrix is usually singular, we improve the method by performing the pseudoinversion by singular value decomposition, keeping only the four largest singular values. We demonstrate the calibration technique using an imaging polarimeter based upon liquid crystal variable retarders and with light emitting diode (LED) illumination centered at 472 nm, 525 nm, and 630 nm. We generate the reference polarization states by using an unpolarized source, a single polarizer, and a Fresnel rhomb. This method is particularly useful when calibrations are performed on field-grade instruments at a centrally maintained facility and when a traceability chain needs to be maintained.     

Stochastic analysis of biochemical reaction networks with absolute concentration robustness

It has recently been shown that structural conditions on the reaction network, rather than a ‘fine-tuning’ of system parameters, often suffice to impart ‘absolute concentration robustness’ (ACR) on a wide class of biologically relevant, deterministically modelled mass-action systems. We show here that fundamentally different conclusions about the long-term behaviour of such systems are reached if the systems are instead modelled with stochastic dynamics and a discrete state space. Specifically, we characterize a large class of models that exhibit convergence to a positive robust equilibrium in the deterministic setting, whereas trajectories of the corresponding stochastic models are necessarily absorbed by a set of states that reside on the boundary of the state space, i.e. the system undergoes an extinction event. If the time to extinction is large relative to the relevant timescales of the system, the process will appear to settle down to a stationary distribution long before the inevitable extinction will occur. This quasi-stationary distribution is considered for two systems taken from the literature, and results consistent with ACR are recovered by showing that the quasi-stationary distribution of the robust species approaches a Poisson distribution.     

Quoting manufacturing due dates subject to a service level constraint

We develop a method for quoting manufacturing due dates to achieve a target service level (percent of orders filled on-time). We posit a very general function for determining leadtimes as a function of work in process and use a control chart method for adjusting the parameters in this function over time. We make almost no assumptions about the nature of the underlying production system (i.e., we do not require any particular distribution of process times, nor do we require that the system be in steady state). Using simulation we show that our method is very accurate for a simple case where an exact analytic solution is possible and that it outperforms other due date quoting methods from the literature in more complex situations.     

Space and contact networks: capturing the locality of disease transmission.

While an arbitrary level of complexity may be included in simulations of spatial epidemics, computational intensity and analytical intractability mean that such models often lack transparency into the determinants of epidemiological dynamics. Although numerous approaches attempt to resolve this complexity-tractability trade-off, moment closure methods arguably offer the most promising and robust frameworks for capturing the role of the locality of contact processes on global disease dynamics. While a close analogy may be made between full stochastic spatial transmission models and dynamic network models, we consider here the special case where the dynamics of the network topology change on time-scales much longer than the epidemiological processes imposed on them; in such cases, the use of static network models are justified. We show that in such cases, static network models may provide excellent approximations to the underlying spatial contact process through an appropriate choice of the effective neighbourhood size. We also demonstrate the robustness of this mapping by examining the equivalence of deterministic approximations to the full spatial and network models derived under third-order moment closure assumptions. For systems where deviation from homogeneous mixing is limited, we show that pair equations developed for network models are at least as good an approximation to the underlying stochastic spatial model as more complex spatial moment equations, with both classes of approximation becoming less accurate only for highly localized kernels.     

The diminishing role of hubs in dynamical processes on complex networks

It is notoriously difficult to predict the behaviour of a complex self-organizing system, where the interactions among dynamical units form a heterogeneous topology. Even if the dynamics of each microscopic unit is known, a real understanding of their contributions to the macroscopic system behaviour is still lacking. Here, we develop information-theoretical methods to distinguish the contribution of each individual unit to the collective out-of-equilibrium dynamics. We show that for a system of units connected by a network of interaction potentials with an arbitrary degree distribution, highly connected units have less impact on the system dynamics when compared with intermediately connected units. In an equilibrium setting, the hubs are often found to dictate the long-term behaviour. However, we find both analytically and experimentally that the instantaneous states of these units have a short-lasting effect on the state trajectory of the entire system. We present qualitative evidence of this phenomenon from empirical findings about a social network of product recommendations, a protein–protein interaction network and a neural network, suggesting that it might indeed be a widespread property in nature.     

Availability maximization under partial observations

In this paper, we propose a new model for availability maximization under partial observations for maintenance applications. Compared with the widely studied cost minimization models, few structural results are known about the form of the optimal control policy for availability maximization models. We consider a failing system with unobservable operational states. Only the failure state is observable. System deterioration is driven by an unobservable, continuous-time homogeneous Markov process. Multivariate condition monitoring data which is stochastically related to the unobservable state of the system is collected at equidistant sampling epochs and is used to update the posterior state distribution for decision making. Preventive maintenance can be carried out at any sampling epoch, and corrective maintenance is carried out upon system failure. The objective is to determine the form of the optimal control policy that maximizes the long-run expected average availability per unit time. We formulate the problem as an optimal stopping problem with partial information. Under standard assumptions, we prove that a control limit policy is optimal. A computational algorithm is developed, illustrated by numerical results.     

Finite energy states generation by periodically pulsed nonlinear oscillator

Abstract

A system comprising an anharmonic oscillator interacting with an external filed is discussed. The oscillator is assumed to be driven by a series of ultra-short external coherent pulses, and is initially in the vacuum state. It is shown that despite strong and permanent external excitations and lack of damping processes, for sufficiently short times between two subsequent pulses the system's evolution is restricted to a finite set of Fock n-photon states. Consequently, the mean energy of photons remains finite and the system evolves similarly to displaced Kerr states. The dynamics of the mean energy of photons is compared to that of its classical counterparts.     

Resonant tunneling in normal metal-superconductor junctions: Low-voltage conductance features and excess current

We have studied the conductance of normal metal-superconductor junctions with a potential barrier on the interface containing localized electron states. The probabilities of the Andreev reflection and the quasiparticle transmission are calculated using a one-dimensional model with a single localized state. The conductance of the junctions with many localized states described by a uniform distribution of the bound state energies around the Fermi level is calculated in the low-temperature region. It is shown that the Andreev reflection processes assisted by the resonant tunneling result in new features of the conductance, such as a zerobias maximum, a low-energy gaplike structure, and a two-gap behavior of the conductance at applied voltages near the value corresponding to the energy gap of the superconductor. The current-voltage characteristics for high applied voltages reveal both an excess current and current deficit, depending on the parameters of the localized states and the transparency of the tunnel barrier.     

Embedded discontinuity finite element method for modeling of localized failure in heterogeneous materials with structured mesh: an alternative to extended finite element method

In this work we discuss the finite element model using the embedded discontinuity of the strain and displacement field, for dealing with a problem of localized failure in heterogeneous materials by using a structured finite element mesh. On the chosen 1D model problem we develop all the pertinent details of such a finite element approximation. We demonstrate the presented model capabilities for representing not only failure states typical of a slender structure, with crack-induced failure in an elastic structure, but also the failure state of a massive structure, with combined diffuse (process zone) and localized cracking. A robust operator split solution procedure is developed for the present model taking into account the subtle difference between the types of discontinuities, where the strain discontinuity iteration is handled within global loop for computing the nodal displacement, while the displacement discontinuity iteration is carried out within a local, element-wise computation, carried out in parallel with the Gauss-point computations of the plastic strains and hardening variables. The robust performance of the proposed solution procedure is illustrated by a couple of numerical examples. Concluding remarks are stated regarding the class of problems where embedded discontinuity finite element method (ED-FEM) can be used as a favorite choice with respect to extended FEM (X-FEM).     

Noise classification and automatic restoration system using non-local regularization frameworks

ABSTRACT

Medical, satellite or microscopic images differ in the imaging techniques used, hence their underlying noise distribution also are different. Most of the restoration methods including regularization models make prior assumptions about the noise to perform an efficient restoration. Here we propose a system that estimates and classifies the noise into different distributions by extracting the relevant features. The system provides information about the noise distribution and then it gets directed into the restoration module where an appropriate regularization method (based on the non-local framework) has been employed to provide an efficient restoration of the data. We have effectively addressed the distortion due to data-dependent noise distributions such as Poisson and Gamma along with data uncorrelated Gaussian noise. The studies have shown a 97.7% accuracy in classifying noise in the test data. Moreover, the system also shows the capability to cater to other popular noise distributions such as Rayleigh, Chi, etc.     

Superposition states of light in a three-photon absorbing dissipative medium

We consider the process of simultaneous absorption of three photons in a medium in the presence of weak one-photon absorption. We show that in such a system stationary three-component superposition states of light may be formed in the range of small values of state amplitude (weak perturbation). This circumstance is associated with the fact that in this range of interaction the field spends more time in one of the three types of superposition states of light (constituting an ensemble of quantum trajectories of the system) than in two other types. We also show that in the range of large values of state amplitude it is possible to obtain three types of non-stationary superposition states of light. By using numerical simulation of quantum trajectories of the system we study the dynamics of the quantum entropy of the field. We calculate the Wigner functions of the field states. We also obtain analytical results for the density matrix of the steady state of the system.     

Optical state measurement with a two-component probe

Abstract

A state of light which is a superposition of the vacuum and the one-photon number state is the simplest state containing phase information. Recently we have shown how a field in such a state might be generated and here we explore its usefulness as a probe for measuring unknown states of light. We find that this probe can be used reasonably simply both to determine completely some pure states of light and to measure the diagonal and nearest off-diagonal elements of the density matrix in the number state basis and hence to obtain the mean sine and cosine of the phase of an unknown mixed state. We suggest further how a field in a superposition of the vacuum and the two-photon number state might be generated and how this can be used as a probe, both to measure the off-diagonal matrix elements second nearest to the diagonal of a mixed state density matrix and to measure the variance of the cosine and the sine of the phase. We also examine the experimentally more likely case where the probe fields are in mixed states and show how the same information about the unknown state can still be retrieved.     

A stochastic regime switching model for the failure process of a repairable system

This paper presents a stochastic model and estimation procedure for analyzing the failure process of a repairable system. We consider repairable systems whose successive interfailure times reveal a significant dependence while showing an insignificant trend. Neither the renewal process nor the non-homogeneous Poisson process are adequate for modeling such failure processes. Especially when the interfailure times show a cyclic pattern, we may consider a switching of the regimes (states) governing the lifetime distribution of the system. We propose a Markov switching model describing the failure process for such a case. The model postulates that a finite number of states governs the distinct lifetime distributions, and the state makes transitions according to a discrete-time Markov chain. Each of the distinct lifetime distributions represents a failure type that may change after successive repairs. Our model generalizes the mixture model by allowing the mixture probabilities to change during the transient period of the system. The model can capture the transient behavior of the system. The interfailure times constitute a set of incomplete data because the states are not explicitly identified. For the incomplete data, we propose a procedure for finding the maximum likelihood estimates of the model parameters by adopting the expectation and maximization principle. We also suggest a statistical method to determine the number of significant states. A Monte Carlo study is performed with two-parameter Weibull lifetime distributions. The results show consistency and good properties of the estimates. Some sets of Proschan's air conditioning unit data [Technometrics, 1963, 5′ 375–383] are also analyzed and the results are discussed with respect to the number of significant states and the performance of the prediction.     

Delayed self-regulation and time-dependent chemical drive leads to novel states in epigenetic landscapes

The epigenetic pathway of a cell as it differentiates from a stem cell state to a mature lineage-committed one has been historically understood in terms of Waddington''s landscape, consisting of hills and valleys. The smooth top and valley-strewn bottom of the hill represent their undifferentiated and differentiated states, respectively. Although mathematical ideas rooted in nonlinear dynamics and bifurcation theory have been used to quantify this picture, the importance of time delays arising from multistep chemical reactions or cellular shape transformations have been ignored so far. We argue that this feature is crucial in understanding cell differentiation and explore the role of time delay in a model of a single-gene regulatory circuit. We show that the interplay of time-dependent drive and delay introduces a new regime where the system shows sustained oscillations between the two admissible steady states. We interpret these results in the light of recent perplexing experiments on inducing the pluripotent state in mouse somatic cells. We also comment on how such an oscillatory state can provide a framework for understanding more general feedback circuits in cell development.     

Precise, controlled laser delivery with evanescent optical waves

Precise laser surgery is possible with laser pulses at wavelengths that are strongly absorbed at the surface of tissue. However, pulses at these wavelengths (far UV, far infrared) are not compatible with fiber-optic transmission, making endoscopic surgical procedures inside the body difficult. We use evanescent optical waves to demonstrate an alternative for confining energy near the tissue surface. Precise, superficial tissue ablation is achieved with evanescent waves generated at a sapphire-tissue interface by a free-electron laser, where the ablation depth may be varied. A new class of precise, controlled laser surgical tools may be achieved in this novel approach for use in endoscopic procedures. Electromagnetic theory governing evanescent-wave tissue ablation is presented.     

Squeezed States in Non-ideal Interferometers: The Effect of Aberrations

Abstract

We extend a previous study of the usefulness of squeezed states of light for noise reduction in a non-ideal interferometer to the case where fringe visibility that is less than unity is caused by non-absorptive processes such as aberrations or misalignment of the beams. The main differences and similarities between this system and our earlier study of an interferometer with unbalanced losses in both arms are presented and discussed. An expression is given for the minimum detectable phase change in this system, in the presence of squeezing, as a function of fringe visibility.