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We develop methods to find the limits to finite-time single photon extraction from emitter-cavity systems. We first establish analytic upper and lower bounds on the maximum extraction probability from a canonical $\Lambda$-system before developing a numeric method to optimise generic output probabilities from $\Lambda$-systems generalised to multiple ground states. We use these methods to study the limits to finite-time photon extraction and the wavepackets that satisfy them, finding that using an optimised wavepacket ranging between a sinusoidal and exponentially decaying profile can considerably reduce photon duration for a given extraction efficiency. We further optimise the rates of quantum protocols requiring emitter-photon correlation to obtain driving-independent conclusions about the effect of system parameters on success probability. We believe that these results and methods will provide valuable tools and insights for the development of cavity-based single photon sources combining high efficiency and high rate., Comment: 26 pages, 7 figures

The transverse field structure and diffraction loss of the resonant modes of Fabry-P\'erot optical cavities are acutely sensitive to the alignment and shape of the mirror substrates. We develop extensions to the `mode mixing' method applicable to arbitrary mirror shapes, which both facilitate fast calculation of the modes of cavities with transversely misaligned mirrors and enable the determination and transformation of the geometric properties of these modes. We show how these methods extend previous capabilities by including the practically-motivated case of transverse mirror misalignment, unveiling rich and complex structure of the resonant modes., Comment: 17 pages, 7 figures

We present a study on the optical losses of Fabry-P\'erot cavities subject to realistic transverse mirror misalignment. We consider mirrors of the two most prevalent surface forms: idealised spherical depressions, and Gaussian profiles generated by laser ablation. We first describe the mode mixing phenomena seen in the spherical mirror case and compare to the frequently-used clipping model, observing close agreement in the predicted diffraction loss, but with the addition of protective mode mixing at transverse degeneracies. We then discuss the Gaussian mirror case, detailing how the varying surface curvature across the mirror leads to complex variations in round trip loss and mode profile. In light of the severe mode distortion and strongly elevated loss predicted for many cavity lengths and transverse alignments when using Gaussian mirrors, we suggest that the consequences of mirror surface profile are carefully considered when designing cavity experiments., Comment: 20 pages, 12 figures

We demonstrate the use of machine learning through convolutional neural networks to solve inverse design problems of optical resonator engineering. The neural network finds a harmonic modulation of a spherical mirror to generate a resonator mode with a given target topology ("mode on-demand"). The procedure allows us to optimize the shape of mirrors to achieve a significantly enhanced coupling strength and cooperativity between a resonator photon and a quantum emitter located at the center of the resonator. In a second example, a double-peak mode is designed which would enhance the interaction between two quantum emitters, e.g., for quantum information processing., Comment: 9 pages, 7 figures

We propose a procedure for the significant enhancement of the strong coupling rate between photons in an optical cavity and a single quantum emitter, such as an atom, quantum dot or trapped ion. We show that specially designed, non-spherical mirrors can lead to cavity eigenmodes that exhibit a large field enhancement at the center of the cavity while inducing significantly less beam divergence, and therefore smaller round trip losses and higher cooperativity, than can be achieved by operating a spherical-mirror cavity in the near-concentric regime. We verify our designs using mode matching theory and discuss their robustness relative to different kinds of manufacturing deviations., Comment: 9 pages, 7 figures

The effect of slow light on second harmonic generation in a periodically poled $\chi^{(2)}$ nonlinear medium is investigated theoretically. A linear $\pi$ phase shifted grating is used to slow the group velocity of the fundamental frequency and the resulting field enhancement greatly increases the second harmonic conversion efficiency. A second linear grating at the input end ensures that all output is in the forward direction. We show that almost 100\% conversion efficiency can be achieved for continuous wave pumping at low intensities that generate negligible conversion in the absence of the slow-light grating., Comment: 8 pages, 17 figures

We investigate a theoretical model for a dynamic Moir\'e grating which is capable of producing slow and stopped light with improved performance when compared with a static Moir\'e grating. A Moir\'e grating superimposes two grating periods which creates a narrow slow light resonance between two band gaps. A Moir\'e grating can be made dynamic by varying its coupling strength in time. By increasing the coupling strength the reduction in group velocity in the slow light resonance can be improved by many orders of magnitude while still maintaining the wide bandwidth of the initial, weak grating. We show that for a pulse propagating through the grating this is a consequence of altering the pulse spectrum and therefore the grating can also perform bandwidth modulation. Finally we present a possible realization of the system via an electro-optic grating by applying a quasi-static electric field to a poled $\chi^{(2)}$ nonlinear medium., Comment: 9 pages, 14 figures

It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well. In addition, it is shown that the number of perfect matchings in any cubic graph $G$ equals the expected value of a random variable defined on all $2$-colorings of edges of $G$. Finally, an improved lower bound on the maximum number of cycles in a cubic graph is provided., Comment: 20 pages, 19 figures

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given., Comment: 28 pages, 2 figures

We theoretically and numerically investigate intermodal four-wave-mixing in km-long fibers, where random birefringence fluctuations are present along the fiber length. We identify several distinct regimes that depend on the relative magnitude between the length scale of the random fluctuations and the beat-lengths of the interacting quasi-degenerate modes. In addition, we analyze the impact of polarization mode-dispersion and we demonstrate that random variations of the core radius, which are typically encountered during the drawing stage of the fiber, can represent the major source of bandwidth impairment. These results set a boundary on the limits of validity of the classical Manakov model and may be useful for the design of multimode parametric amplifiers and wavelength converters, as well as for the analysis of nonlinear impairments in long-haul spatial division multiplexed transmission.

In fiber based Fabry-P\'{e}rot Cavities (FFPCs), limited spatial mode matching between the cavity mode and input/output modes has been the main hindrance for many applications. We have demonstrated a versatile mode matching method for FFPCs. Our novel design employs an assembly of a graded-index and large core multimode fiber directly spliced to a single mode fiber. This all-fiber assembly transforms the propagating mode of the single mode fiber to match with the mode of a FFPC. As a result, we have measured a mode matching of 90\% for a cavity length of $\sim$400 $\mu m$. This is a significant improvement compared to conventional FFPCs coupled with just a single mode fiber, especially at long cavity lengths. Adjusting the parameters of the assembly, the fundamental cavity mode can be matched with the mode of almost any single mode fiber, making this approach highly versatile and integrable., Comment: 6 pages, 5 figures, article

We study a range of radio-frequency ion trap geometries and investigate the effect of integrating dielectric cavity mirrors on their trapping potential. We aim to identify ion trap and cavity configurations that are best suited for achieving small cavity volumes and thus large ion-photon coupling as required for scalable quantum information networks. In particular, we investigate the trapping potential distortions caused by the dielectric material of the cavity mirrors for different mirror orientations with respect to the trapping electrodes, as well as for mirror misalignment. We also analyze the effect of the mirror material properties such as dielectric constants and surface conductivity, and study the effect of surface charges on the mirrors. The smallest trapping potential distortions are found if the cavities are aligned along the major symmetry axis of the electrode geometries. These cavity configurations also appear to be the most stable with respect to any mirror misalignment.

In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert's Nullstellensatz we obtain a necessary condition for tiling $n$-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of $n$-space by translates of a cluster $V$ of prime size then there is a lattice tiling by $V$ as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster $V$ is lattice if $V$ generates $n$-space.

Nowadays sparse systems of equations occur frequently in science and engineering. In this contribution we deal with sparse systems common in cryptanalysis. Given a cipher system, one converts it into a system of sparse equations, and then the system is solved to retrieve either a key or a plaintext. Raddum and Semaev proposed new methods for solving such sparse systems. It turns out that a combinatorial MaxMinMax problem provides bounds on the average computational complexity of sparse systems. In this paper we initiate a study of a linear algebra variation of this MaxMinMax problem.

We investigate the subwavelength imaging capacity of a two-dimensional fanned-out plasmonic waveguide array, formed by air channels surrounded by gold metal layers for operation at near-infrared wavelengths, via finite element simulations. High resolution is achieved on one side of the device by tapering down the channel width while simultaneously maintaining propagation losses of a few dB. On the other, low-resolution side, output couplers are designed to optimize coupling to free space and to minimize channel cross talk via surface plasmons. Point sources separated by {\lambda}/15 can still be clearly distinguished. Moreover, up two 90% of the power of a point dipole is coupled to the device. Applications are high-resolution linear detector arrays and, by operating the device in reverse, high-resolution optical writing.

The "Gluing Algorithm" of Semaev [Des.\ Codes Cryptogr.\ 49 (2008), 47--60] --- that finds all solutions of a sparse system of linear equations over the Galois field $GF(q)$ --- has average running time $O(mq^{\max \left\vert \cup_{1}^{k}X_{j}\right\vert -k}), $ where $m$ is the total number of equations, and $\cup_{1}^{k}X_{j}$ is the set of all unknowns actively occurring in the first $k$ equations. Our goal here is to minimize the exponent of $q$ in the case where every equation contains at most three unknowns. %Applying hypergraph-theoretic methods we prove The main result states that if the total number $\left\vert \cup_{1}^{m}X_{j}\right\vert$ of unknowns is equal to $m$, then the best achievable exponent is between $c_1m$ and $c_2m$ for some positive constants $c_1$ and $c_2.$

Motivated by a problem in computer architecture we introduce a notion of the perfect distance-dominating set, PDDS, in a graph. PDDSs constitute a generalization of perfect Lee codes, diameter perfect codes, as well as other codes and dominating sets. In this paper we initiate a systematic study of PDDSs. PDDSs related to the application will be constructed and the non-existence of some PDDSs will be shown. In addition, an extension of the long-standing Golomb-Welch conjecture, in terms of PDDS, will be stated. We note that all constructed PDDSs are lattice-like which is a very important feature from the practical point of view as in this case decoding algorithms tend to be much simpler., Comment: 17 pages, 13 figures; Designs, Codes and Cryptography 2012

An $n$-dimensional cross comprises $2n+1$ unit cubes: the center cube and reflections in all its faces. It is well known that there is a tiling of $R^{n}$ by crosses for all $n.$ AlBdaiwi and the first author proved that if $2n+1$ is not a prime then there are $2^{\aleph_{0}}$ \ non-congruent regular (= face-to-face) tilings of $R^{n}$ by crosses, while there is a unique tiling of $R^{n}$ by crosses for $n=2,3$. They conjectured that this is always the case if $2n+1$ is a prime. To support the conjecture we prove in this paper that also for $R^{5}$ there is a unique regular, and no non-regular, tiling by crosses. So there is a unique tiling of $R^{3}$ by crosses, there are $2^{\aleph_{0}}$ tilings of $R^{4},$ but for $R^{5}$ there is again only one tiling by crosses. We guess that this result goes against our intuition that suggests "the higher the dimension of the \ space, the more freedom we get".

The Golomb-Welch conjecture deals with the existence of perfect $e$% -error correcting Lee codes of word length $n,$ $PL(n,e)$ codes. Although there are many papers on the topic, the conjecture is still far from being solved. In this paper we initiate the study of an invariant connected to abelian groups that enables us to reformulate the conjecture, and then to prove the non-existence of linear PL(n,2) codes for $n\leq 12$. Using this new approach we also construct the first quasi-perfect Lee codes for dimension $n=3,$ and show that, for fixed $n$, there are only finitely many such codes over $Z^n$.

Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all $q$ for which there exists a linear diameter-4 perfect Lee code of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This is in a strict contrast with perfect error-correcting Lee codes of word length $n$ over $Z\,$\ as there is a unique such code for $n=3,$ and its is conjectured that this is always the case when $2n+1$ is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper.

We compare the efficiencies of two optical cooling schemes, where a single particle is either inside or outside an optical cavity, under experimentally-realisable conditions. We evaluate the cooling forces using the general solution of a transfer matrix method for a moving scatterer inside a general one-dimensional system composed of immobile optical elements. Assuming the same atomic saturation parameter, we find that the two cooling schemes provide cooling forces and equilibrium temperatures of comparable magnitude., Comment: 6 pages, 6 figures