The Casimir force and the quantum theory of lossy optical cavities

We present a new derivation of the Casimir force between two parallel plane mirrors at zero temperature. The two mirrors and the cavity they enclose are treated as quantum optical networks. They are in general lossy and characterized by frequency dependent reflection amplitudes. The additional fluctuations accompanying losses are deduced from expressions of the optical theorem. A general proof is given for the theorem relating the spectral density inside the cavity to the reflection amplitudes seen by the inner fields. This density determines the vacuum radiation pressure and, therefore, the Casimir force. The force is obtained as an integral over the real frequencies, including the contribution of evanescent waves besides that of ordinary waves, and, then, as an integral over imaginary frequencies. The demonstration relies only on general properties obeyed by real mirrors which also enforce general constraints for the variation of the Casimir force.Comment: 18 pages, 6 figures, minor amendment

PT-symmetric cross injection dual optoelectronic oscillator

An optoelectronic oscillator (OEO) is a time delay oscillator (TDO) that uses photonics technology to provide the long delay required to generate pristine microwave carriers. Parity-time (PT) symmetry concepts applied to an OEO offer the potential to achieve combined low phase noise and high sidemode suppression. A TDO composed of a pair of identical ring resonators coupled by a 2x2 coupler is modelled, and the coupler transmission matrix required for the oscillator to be PT- symmetric is derived. In a first configuration, the coupler is interpreted as the composition of a gain/loss block and a Mach-Zehnder interferometer (MZI) block. In practice, there are excess losses that must be compensated by a special dual amplifier with saturation characteristics compatible with PT- symmetry. The PT- symmetry phase transition determined by the gain/loss and the MZI differential phase parameters is found to be global and not local in its effect on modes. This is resolved by placing a short delay line within one arm of the MZI resulting in a frequency dependent and hence local mode-selective PT- symmetry phase transition. In addition, it is demonstrated that the first configuration may be transformed into a second but equivalent configuration as a cross-injection dual TDO with imbalanced delays. The local PT- symmetry phase transition may then be understood in terms of the Vernier effect. Advantageously, the second configuration enables the special dual amplifier to be replaced by a pair of matched but otherwise independent amplifiers. Thereby, the second configuration is amenable to practical implementation as a dual OEO using standard RF-photonic and RF-electronic components. The theory is validated by complex envelope model simulations using Simulink and phase model analytic results evaluated using MATLAB. There is excellent agreement between the theoretical and simulation results.Comment: 40 pages, 13 figure

Linear stochastic systems: a white noise approach

Using the white noise setting, in particular the Wick product, the Hermite transform, and the Kondratiev space, we present a new approach to study linear stochastic systems, where randomness is also included in the transfer function. We prove BIBO type stability theorems for these systems, both in the discrete and continuous time cases. We also consider the case of dissipative systems for both discrete and continuous time systems. We further study $\ell_1$-$\ell_2$ stability in the discrete time case, and ${\mathbf L}_2$-${\mathbf L}_\infty$ stability in the continuous time case

A group-theoretic approach to formalizing bootstrapping problems

The bootstrapping problem consists in designing agents that learn a model of themselves and the world, and utilize it to achieve useful tasks. It is different from other learning problems as the agent starts with uninterpreted observations and commands, and with minimal prior information about the world. In this paper, we give a mathematical formalization of this aspect of the problem. We argue that the vague constraint of having "no prior information" can be recast as a precise algebraic condition on the agent: that its behavior is invariant to particular classes of nuisances on the world, which we show can be well represented by actions of groups (diffeomorphisms, permutations, linear transformations) on observations and commands. We then introduce the class of bilinear gradient dynamics sensors (BGDS) as a candidate for learning generic robotic sensorimotor cascades. We show how framing the problem as rejection of group nuisances allows a compact and modular analysis of typical preprocessing stages, such as learning the topology of the sensors. We demonstrate learning and using such models on real-world range-finder and camera data from publicly available datasets

Duality Symmetry

Symmetry is one of the most general concepts in physics. Symmetry arguments are used to explain and predict observations at all length scales, from elementary particles to cosmology. The generality of symmetry arguments, combined with their simplicity, makes them a powerful tool for both fundamental and applied investigations. In electrodynamics, one of the symmetries is the invariance of the equations under exchange of electric and magnetic quantities. The continuous version of this symmetry is most commonly known as electromagnetic duality symmetry. This concept has been accepted for more than a century, and, throughout this time, has influenced other areas of physics, like high energy physics and gravitation. This Special Issue is devoted to electromagnetic duality symmetry and other vareities of dualities in physics. It contains four Articles, one Review and one Perspective. The context of the contributions ranges from string theory to applied nanophotonics, which, as anticipated, shows that duality symmetries in general and electromagnetic duality symmetry in particular are useful in a wide variety of physics fields, both theoretical and applied. Moreover, a number of the contributions show how the use of symmetry arguments and the quantification of symmetry breaking can successfully guide our theoretical understanding and provide us with guidelines for system design