854 research outputs found
Cavity nonlinear optics with few photons and ultracold quantum particles
The light force on particles trapped in the field of a high-Q cavity mode
depends on the quantum state of field and particle. Different photon numbers
generate different optical potentials anddifferent motional states induce
different field evolution. Even for weak saturation and linear polarizability
the induced particle motion leads to nonlinear field dynamics. We derive a
corresponding effective field Hamiltonian containing all the powers of the
photon number operator, which predicts nonlinear phase shifts and squeezing
even at the few-photon level. Wave-function simulations of the full
particle-field dynamics confirm this and show significant particle-field
entanglement in addition.Comment: 5 pages, 5 figure
The topology of Stein fillable manifolds in high dimensions II
We continue our study of contact structures on manifolds of dimension at
least five using complex surgery theory. We show that in each dimension 2q+1 >
3 there are 'maximal' almost contact manifolds to which there is a Stein
cobordism from any other (2q+1)-dimensional contact manifold. We show that the
product M x S^2 admits a weakly fillable contact structure provided M admits a
weak symplectic filling. We also study the connection between Stein fillability
and connected sums: we give examples of almost contact manifolds for which the
connected sum is Stein fillable, while the components are not.
Concerning obstructions to Stein fillings, we show that the
(8k-1)-dimensional sphere has an almost contact structure which is not Stein
fillable once k > 1. As a consequence we deduce that any highly connected
almost contact (8k-1)-manifold (with k > 1) admits an almost contact structure
which is not Stein fillable. The proofs rely on a new number-theoretic result
about Bernoulli numbers.Comment: We corrected mistakes in the proofs of Lemma 2.9 and Corollary 2.10.
This lead to an assumption being removed from the statement of Theorem 1.3.
The paper is now published in Geometry and Topology. The appendix was written
by Bernd C. Kellne
Adequacy of the Dicke model in cavity QED: a counter-"no-go" statement
The long-standing debate whether the phase transition in the Dicke model can
be realized with dipoles in electromagnetic fields is yet an unsettled one. The
well-known statement often referred to as the "no-go theorem", asserts that the
so-called A-square term, just in the vicinity of the critical point, becomes
relevant enough to prevent the system from undergoing a phase transition. At
variance with this common belief, in this paper we prove that the Dicke model
does give a consistent description of the interaction of light field with the
internal excitation of atoms, but in the dipole gauge of quantum
electrodynamics. The phase transition cannot be excluded by principle and a
spontaneous transverse-electric mean field may appear. We point out that the
single-mode approximation is crucial: the proper treatment has to be based on
cavity QED, wherefore we present a systematic derivation of the dipole gauge
inside a perfect Fabry-P\'erot cavity from first principles. Besides the impact
on the debate around the Dicke phase transition, such a cleanup of the
theoretical ground of cavity QED is important because currently there are many
emerging experimental approaches to reach strong or even ultrastrong coupling
between dipoles and photons, which demand a correct treatment of the Dicke
model parameters
Equivariant D-modules on 2x2xn hypermatrices
We study D-modules and related invariants on the space of 2 x 2 x n
hypermatrices for n >= 3, which has finitely many orbits under the action of G
= GL_2 x GL_2 x GL_n. We describe the category of coherent G-equivariant
D-modules as the category of representations of a quiver with relations. We
classify the simple equivariant D-modules, determine their characteristic
cycles and find special representations that appear in their G-structures. We
determine the explicit D-module structure of the local cohomology groups with
supports given by orbit closures. As a consequence, we calculate the Lyubeznik
numbers and intersection cohomology groups of the orbit closures. All but one
of the orbit closures have rational singularities: we use local cohomology to
prove that the one exception is neither normal nor Cohen--Macaulay. While our
results display special behavior in the cases n=3 and n=4, they are completely
uniform for n >= 5.Comment: 45 page
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