Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

Starting from equations obeyed by functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presente

Adiabatic tracking for photo- and magneto-association of Bose-Einstein condensates with Kerr nonlinearities

We develop the method of adiabatic tracking for photo- and magneto-association of Bose-Einstein atomic condensates with models that include Kerr type nonlinearities. We show that the inclusion of these terms can produce qualitatively important modifications in the adiabatic dynamics, like the appearance of bifurcations, in which the trajectory that is being tracked loses its stability. As a consequence the adiabatic theorem does not apply and the adiabatic transfer can be strongly degraded. This degradation can be compensated by using fields that are strong enough compared with the values of the Kerr terms. The main result is that, despite these potentially detrimental features, there is always a choice of the detuning that leads to an efficient adiabatic tracking, even for relatively weak fields

New powerful outburst of the unusual young star V1318 Cyg S (LkH{\alpha} 225)

Young double star V1318 Cyg, which is associated with a small isolated star-forming region around HAeBe star BD+40o4124, has very unusual photometric and spectral behavior. We present results of photometric and spectroscopic observations in the optical range. We carried out BVRI CCD photometric observations of V1318 Cyg from 2015 Sept. to 2017 July. For the same period we acquired medium- and low-resolution spectra. Observations were performed with the 2.6 m telescope. We analyze the historical light curve for V1318 Cyg and demonstrate that the southern component, V1318 Cyg S, after being rather bright in the 1970s (V$\approx$14 mag) started to lower its brightness and in 1990 became practically invisible in the optical. After its reappearance in the second half of the 1990s the star started to become very slowly brighter. Between 2006 and 2010 V1318 Cyg S started brightening more quickly, and in 2015 had become brighter by more than five magnitudes in visible light. Since this time V1318 Cyg S has remained at this maximum. Its spectrum shows little variability and consists of a mixture of emission and absorption lines, which has allowed for estimates of its spectral type as early Ae, with obvious evidence of matter outflow. We derive its current A(V)$\approx$ 7.2 and L = 750 L(sun) thus confirming that V1318 Cyg S should belong to the Herbig Ae stars, making it, along with BD+40o4124 and V1686 Cyg, the third luminous young star in the group. It is very probable that we observe V1318 Cyg S near the pole and that the inclination of its dense and slow ($\approx$ 100 km/s) outflow is low. The unusual variability and other features of V1318 Cyg S make it difficult to classify this star among known types of eruptive young stars. It could be an extreme, higher-mass example of an EXor, or an object of intermediate class between EXors and FUors, like V1647 Ori.Comment: 9 pages, 7 figure

Three-photon states in nonlinear crystal superlattices

It has been a longstanding goal in quantum optics to realize controllable sources generating joint multiphoton states, particularly, photon triplet with arbitrary spectral characteristics. We demonstrate that such sources can be realized via cascaded parametric down-conversion (PDC) in superlattice structures of nonlinear and linear segments. We consider scheme that involves two parametric processes: $\omega_{0}\rightarrow\omega_{1}+\omega_{2}$, $\omega_{2}\rightarrow\omega_{1}+\omega_{1}$ under pulsed pump and investigate spontaneous creation of photon triplet as well as generation of high-intensity mode in intracavity three-photon splitting. We show preparation of Greenberger-Horne-Zeilinger polarization entangled states in cascaded type-II and type-I PDC in framework of consideration dual-grid structure that involves two periodically-poled crystals. We demonstrate the method of compensation of the dispersive effects in non-linear segments by appropriately chosen linear dispersive segments of superlattice for preparation heralded joint states of two polarized photons. In the case of intracavity three-photon splitting, we concentrate on investigation of photon-number distributions, third-order photon-number correlation function as well as the Wigner functions. These quantities are observed both for short interaction time intervals and in over transient regime, when dissipative effects are essential.Comment: 15 pages, 6 figure