The congruence subgroup problem

This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the non-specialists and avoids technical details.Comment: 10 page

A topological realization of the congruence subgroup Kernel A

A number of years ago, Kumar Murty pointed out to me that the computation of the fundamental group of a Hilbert modular surface ([7],IV,${\S}$6), and the computation of the congruence subgroup kernel of SL(2) ([6]) were surprisingly similar. We puzzled over this, in particular over the role of elementary matrices in both computations. We formulated a very general result on the fundamental group of a Satake compactification of a locally symmetric space. This lead to our joint paper [1] with Lizhen Ji and Les Saper on these fundamental groups. Although the results in it were intriguingly similar to the corresponding calculations of the congruence subgroup kernel of the underlying algebraic group in [5], we were not able to demonstrate a direct connection (cf. [1], ${\S}$7). The purpose of this note is to explain such a connection. A covering space is constructed from inverse limits of reductive Borel-Serre compactifications. The congruence subgroup kernel then appears as the group of deck transformations of this covering. The key to this is the computation of the fundamental group in [1]

Continuous monitoring can improve indistinguishability of a single-photon source

A new engineering technique using continuous quantum measurement in conjunction with feed-forward is proposed to improve indistinguishability of a single-photon source. The technique involves continuous monitoring of the state of the emitter, processing the noisy output signal with a simple linear estimation algorithm, and feed forward to control a variable delay at the output. In the weak coupling regime, the information gained by monitoring the state of the emitter is used to reduce the time uncertainty inherent in photon emission from the source, which improves the indistinguishability of the emitted photons.Comment: 4 pages, 4 figure

Modeling Magnetic Anisotropy of Single Chain Magnets in ∣d/J∣≥1|d/J| \geq 1 Regime

Single molecule magnets (SMMs) with single-ion anisotropies $\mathbf d$, comparable to exchange interactions J, between spins have recently been synthesized. In this paper, we provide theoretical insights into the magnetism of such systems. We study spin chains with site spins, s=1, 3/2 and 2 and on-site anisotropy $\mathbf d$ comparable to the exchange constants between the spins. We find that large $\mathbf d$ leads to crossing of the states with different $M_S$ values in the same spin manifold of the $\mathbf d = 0$ limit. For very large $\mathbf d$'s we also find that the $M_S$ states of the higher energy spin states descend below the $M_S$ states of the ground state spin manifold. Total spin in this limit is no longer conserved and describing the molecular anisotropy by the constants $D_M$ and $E_M$ is not possible. However, the total spin of the low-lying large $M_S$ states is very nearly an integer and using this spin value it is possible to construct an effective spin Hamiltonian and compute the molecular magnetic anisotropy constants $D_M$ and $E_M$. We report effect of finite sizes, rotations of site anisotropies and chain dimerization on the effective anisotropy of the spin chains

Modeling Molecular Magnets with Large Exchange and On-Site Anisotropies

Spins in molecular magnets can experience both anisotropic exchange interactions and on-site magnetic anisotropy. In this paper we study the effect of exchange anisotropy on the molecular magnetic anisotropy both with and without on-site anisotropy. When both the anisotropies are small, we find that the axial anisotropy parameter $D_M$ in the effective spin Hamiltonian is the sum of the individual contributions due to exchange and on-site anisotropies. We find that even for axial anisotropy of about $15\%$, the low energy spectrum does not correspond to a single parent spin manifold but has intruders states arising from other parent spin. In this case, the low energy spectrum can not be described by an effective Hamiltonian spanning the parent spin space. We study the magnetic susceptibility, specific heat as a function of temperature and magnetization as a function of applied field to characterize the system in this limit. We find that there is synergy between the two anisotropies, particularly for large systems with higher site spins.Comment: 30 pages, 11 figures and 3 tables. Supporting information included after the main articl

On the congruence subgroup problem

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