Theory of Two-Photon Interactions with Broadband Down-Converted Light and Entangled Photons

When two-photon interactions are induced by down-converted light with a bandwidth that exceeds the pump bandwidth, they can obtain a behavior that is pulse-like temporally, yet spectrally narrow. At low photon fluxes this behavior reflects the time and energy entanglement between the down-converted photons. However, two-photon interactions such as two-photon absorption (TPA) and sum-frequency generation (SFG) can exhibit such a behavior even at high power levels, as long as the final state (i.e. the atomic level in TPA, or the generated light in SFG) is narrowband enough. This behavior does not depend on the squeezing properties of the light, is insensitive to linear losses, and has potential applications. In this paper we describe analytically this behavior for travelling-wave down-conversion with continuous or pulsed pumping, both for high- and low-power regimes. For this we derive a quantum-mechanical expression for the down-converted amplitude generated by an arbitrary pump, and formulate operators that represent various two-photon interactions induced by broadband light. This model is in excellent agreement with experimental results of TPA and SFG with high power down-converted light and with entangled photons [Dayan et al., Phys. Rev. Lett. 93, 023005 (2004), Dayan et al., Phys. Rev. Lett. 94, 043602, (2005), Pe'er et al., Phys. Rev. Lett. 94, 073601 (2005)].Comment: 23 pages, 4 figures, submitted to Phys. Rev.

Cavity ring-up spectroscopy for ultrafast sensing with optical microresonators

Spectroscopy of whispering-gallery mode (WGM) microresonators has become a powerful scientific tool, enabling detection of single viruses, nanoparticles, and even single molecules. Yet the demonstrated timescale of these schemes has been limited so far to milliseconds or more. Here we introduce a novel scheme that is orders of magnitude faster, capable of capturing complete spectral snapshots of WGM resonances at nanosecond timescales: cavity ring-up spectroscopy (CRUS). Based on sharply-rising detuned probe pulses, CRUS combines the sensitivity of heterodyne measurements with the highest possible, transform-limited acquisition rate. As a demonstration we capture spectra of microtoroid resonators at time intervals as short as 16 ns, directly monitoring sub-microsecond dynamics of their optomechanical vibrations, thermorefractive response and Kerr nonlinearity. CRUS holds promise for the study of fast biological processes such as enzyme kinetics, protein folding and light harvesting, with applications in other fields such as cavity QED and pulsed optomechanics.Comment: 6 pages, 4 figure

Quantum lithography by coherent control of classical light pulses

The smallest spot in optical lithography and microscopy is generally limited by diffraction. Quantum lithography, which utilizes interference between groups of N entangled photons, was recently proposed to beat the diffraction limit by a factor N. Here we propose a simple method to obtain N photons interference with classical pulses that excite a narrow multiphoton transition, thus shifting the "quantum weight" from the electromagnetic field to the lithographic material. We show how a practical complete lithographic scheme can be developed and demonstrate the underlying principles experimentally by two-photon interference in atomic Rubidium, to obtain focal spots that beat the diffraction limit by a factor of 2.Comment: 6 pages, 4 figures, Submitted to Opt. Expres

Random walks on tori and normal numbers in self similar sets

We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K \subset \mathbb{R}^d$ is an attractor of a finite iterated function system of $n\geq 2$ maps of the form $x \mapsto D^{-r_i} x + t_i \ (i=1, \ldots, n)$, where $D$ is an expanding $d\times d$ integer matrix, and is the same for all the maps, and $r_{i} \in\mathbb{N}$, under an irrationality condition on the translation parts $t_i$, almost every point in $K$ (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $x\mapsto Dx$ (multiplication mod $\mathbb{Z}^{d}$). In the one-dimensional case, this conclusion amounts to normality to base $D$. Thus for example, almost every point in an irrational dilation of the middle-thirds Cantor set is normal to base 3.Comment: Theorem 5 in the old version about equidistribution of orbits in fractals has been generalized to the case of different linear contractions which are integer powers of the same linear contraction. We also allow the linear contraction to be non-diagonal, thus we allow some self-affine fractals as wel