Joe P. Chen

Quantum theory of cavity-assisted sideband cooling of mechanical motion.

We present a quantum-mechanical theory of the cooling of a cantilever coupled via radiation pressure to an illuminated optical cavity. Applying the quantum noise approach to the fluctuations of the radiation pressure force, we derive the optomechanical cooling rate and the minimum achievable phonon number. We find that reaching the quantum limit of arbitrarily small phonon numbers requires going into the good-cavity (resolved phonon sideband) regime where the cavity linewidth is much smaller than the mechanical frequency and the corresponding cavity detuning. This is in contrast to the common assumption that the mechanical frequency and the cavity detuning should be comparable to the cavity damping.

Periodic billiard orbits of self-similar Sierpiński carpets

Abstract We identify a collection of periodic billiard orbits in a self-similar Sierpinski carpet billiard table Ω ( S a ) . Based on our refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in S a , we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpinski carpet billiard tables Ω ( S a , n ) . The trivial limit of this sequence then constitutes a periodic orbit of Ω ( S a ) . We also determine the corresponding translation surface S ( S a , n ) for each prefractal table Ω ( S a , n ) , and show that the genera { g n } n = 0 ∞ of a sequence of translation surfaces { S ( S a , n ) } n = 0 ∞ increase without bound. Various open questions and possible directions for future research are offered.

Power dissipation in fractal AC circuits

We extend Feynmans analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Using (weak) self-similarity of our fractal structures, we provide algorithms for studying the equilibrium distribution of energy on these circuits. This extends the analysis of self-similar resistance networks introduced by Fukushima, Kigami, Kusuoka, and more recently studied by Strichartz et al.

Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs

Consider the free field on a fractal graph based on a high-dimensional Sierpinski carpet (e.g. the Menger sponge), that is, a centered Gaussian field whose covariance is the Green’s function for simple random walk on the graph. Moreover assume that a “hard wall” is imposed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitouni for the free field on Zd, d≥3, to the fractal setting.

Regularized Laplacian determinants of self-similar fractals

We study the spectral zeta functions of the Laplacian on fractal sets which are locally self-similar fractafolds, in the sense of Strichartz. These functions are known to meromorphically extend to the entire complex plane, and the locations of their poles, sometimes referred to as complex dimensions, are of special interest. We give examples of locally self-similar sets such that their complex dimensions are not on the imaginary axis, which allows us to interpret their Laplacian determinant as the regularized product of their eigenvalues. We then investigate a connection between the logarithm of the determinant of the discrete graph Laplacian and the regularized one.

Wave Equation on One-Dimensional Fractals with Spectral Decimation and the Complex Dynamics of Polynomials

We study the wave equation on one-dimensional self-similar fractal structures that can be analyzed by the spectral decimation method. We develop efficient numerical approximation techniques and also provide uniform estimates obtained by analytical methods.

From Non-symmetric Particle Systems to Non-linear PDEs on Fractals

We present new results and challenges in obtaining hydrodynamic limits for non-symmetric (weakly asymmetric) particle systems (exclusion processes on pre-fractal graphs) converging to a non-linear heat equation. We discuss a joint density-current law of large numbers and a corresponding large deviations principle.

Quantum Theory of Cavity-Assisted Cantilever Cooling

We present the quantum theory of optomechanical cooling, for a cantilever coupled to an optical cavity. The cantilevers ground state may be reached if the mechanical frequency is larger than the cavity decay rate.