On inverse mean curvature flow in Schwarzschild space and Kottler space

In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $\Sigma$ is strictly mean convex and star-shaped, then the flow hypersurface $\Sigma_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler-Schwarzchild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can use an approximation argument to show the global existence and regularity of the smooth inverse mean curvature flow for star-shaped and weakly mean convex initial hypersurface, which generalizes Huisken-Ilmanen's result [18].Comment: 23 pages, v2, title changed, new result adde

Strongly correlated two-photon transport in a one-dimensional waveguide coupled to a weakly nonlinear cavity

We study the photon-photon correlation properties of two-photon transport in a one-dimensional waveguide coupled to a nonlinear cavity via a real-space approach. It is shown that the intrinsic dissipation of the nonlinear cavity has an important effect upon the correlation of the transported photons. More importantly, strongly correlated photons can be obtained in the transmitted photons even when the nonlinear interaction strength is weak in the cavity. The strong photon-photon correlation is induced by the Fano resonance involving destructive interference between the plane wave and bound state for two-photon transport.Comment: 7 pages, 5 figure

Tunable photon statistics in weakly nonlinear photonic molecules

In recent studies [Liew et al., Phys. Rev. Lett. 104, 183601 (2010); Bamba et al., Phys. Rev. A 83, 021802(R) (2011)], due to destructive interference between different paths for two-photon excitation, strong photon antibunching can be obtained in a photonic molecule consisting of two coupled cavity modes with weak Kerr nonlinearity when one of the cavity modes is driven resonantly. Here, we study the photon statistics in a nonlinear photonic molecule with both the two cavity modes being driven coherently. We show that the statistical properties of the photons can be controlled by regulating the coupling constant between the cavity modes, the strength ratio and the relative phase between the driving fields. The photonic molecules with two driven modes can be used to generate tunable single-photon sources or controlled photonic quantum gates with weak Kerr nonlinearity.Comment: 6 pages, 5 figure

Strong photon antibunching of symmetric and antisymmetric modes in weakly nonlinear photonic molecules

We study the photon statistics of symmetric and antisymmetric modes in a photonic molecule consisting of two linearly coupled nonlinear cavity modes. Our calculations show that strong photon antibunching of both symmetric and antisymmetric modes can be obtained even when the nonlinearity in the photonic molecule is weak. The strong antibunching effect results from the destructive interference between different paths for two-photon excitation. Moreover, we find that the optimal frequency detunings for strong photon antibunching in the symmetric and antisymmetric modes are linearly dependent on the coupling strength between the cavity modes in the photonic molecule. This implies that the photonic molecules can be used to generate tunable single-photon sources by tuning the values of the coupling strength between the cavity modes with weak nonlinearity.Comment: 6 pages, 8 figure

Controllable optical output fields from an optomechanical system with a mechanical driving

We investigate the properties of the optical output fields from a cavity optomechanical system, where the cavity is driven by a strong coupling and a weak probe optical fields and the mechanical resonator is driven by a coherent mechanical pump. When the frequency of the mechanical pump matches the frequency difference between the coupling and probe optical fields, due to the interference between the different optical components at the same frequency, we demonstrate that the large positive or negative group delay of the output field at the frequency of probe field can be achieved and tuned by adjusting the phase and amplitude of the mechanical driving field. Moreover, the strength of the output field at the frequency of optical four-wave-mixing (FWM) field also can be controlled (enhanced and suppressed) by tuning the phase and amplitude of the mechanical pump. We show that the power of the output field at the frequency of the optical FWM field can be suppressed to zero or enhanced so much that it can be comparable with and even larger than the power of the input probe optical field.Comment: 7 pages, 9 figures. Comments are welcome

Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems

We demonstrate the possibility of optical nonreciprocal response in a three-mode optomechanical system where one mechanical mode is optomechanically coupled to two linearly coupled optical modes simultaneously. The optical nonreciprocal behavior is induced by the phase difference between the two optomechanical coupling rates which breaks the time-reversal symmetry of the three-mode optomechanical system. Moreover, the three-mode optomechanical system can also be used as a three-port circulator for two optical and one mechanical modes, which we refer to as optomechanical circulator.Comment: 8 pages, 7 figure

A geometric inequality on hypersurface in hyperbolic space

In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.Comment: 9 pages. All comments are welcom

F\mathcal{F}-stability for self-shrinking solutions to mean curvature flow

In this paper, we formulate the notion of the $\mathcal{F}$-stability of self-shrinking solutions to mean curvature flow in arbitrary codimension. Then we give some classifications of the $\mathcal{F}$-stable self-shrinkers in arbitrary codimension, in codimension one case, our results reduce to Colding-Minicozzi's results.Comment: 23 page

The Gauss-Bonnet-Chern mass for graphic manifolds

In this paper, we prove a positive mass theorem and Penrose-type inequality of the Gauss-Bonnet-Chern mass $m_2$ for the graphic manifold with flat normal bundle.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1304.3504 by other author

Effect of atomic distribution on cooperative spontaneous emission

We study cooperative single-photon spontaneous emission from N multilevel atoms for different atomic distributions in optical vector theory. Instead of the average approximation for interatomic distance or the continuum approximation (sums over atoms replaced by integrals) for atomic distribution, the positions of every atom are taken into account by numerical calculation. It is shown that the regularity of atomic distribution has considerable influence on cooperative spontaneous emission. For a small atomic sample (compared with radiation wavelength), to obtain strong superradiance not only needs the uniform excitation (the Dicke state) but also requires the uniform atomic distribution. For a large sample, the uniform atomic distribution is beneficial to subradiance of the Dicke state, while the influence of atomic distribution on the timed Dicke state is weak and its time evolution obeys exponential decay approximately. In addition, we also investigate the corresponding emission spectrum and verify the directed emission for the timed Dicke state for a large atomic sample.Comment: 9 pages, 9 figure