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Register a new userProjective Hilbert space structures at exceptional points
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A non-Hermitian complex symmetric 2x2 matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behavior in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase jump behavior are analyzed and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT-symmetrically extended Quantum Mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.
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We discuss transformations generated by dynamical quantum systems which are bi-unitary, i.e. unitary with respect to a pair of Hermitian structures on an infinite-dimensional complex Hilbert space. We introduce the notion of Hermitian structures in generic relative position. We provide few necessary and sufficient conditions for two Hermitian structures to be in generic relative position to better illustrate the relevance of this notion. The group of bi-unitary transformations is considered in both the generic and non-generic case. Finally, we generalize the analysis to real Hilbert spaces and extend to infinite dimensions results already available in the framework of finite-dimensional linear bi-Hamiltonian systems.
The defining characteristic of an exceptional point (EP) in the parameter space of a family of operators is that upon encircling the EP eigenstates are permuted. In case one encircles multiple EPs, the question arises how to properly compose the effects of the individual EPs. This was thought to be ambiguous. We show that one can solve this problem by considering based loops and their deformations. The theory of fundamental groups allows to generalize this technique to arbitrary degeneracy structures like exceptional lines in a three-dimensional parameter space. As permutations of three or more objects form a non-abelian group, the next question that arises is whether one can experimentally demonstrate this non-commutative behavior. This requires at least two EPs of a family of operators that have at least 3 eigenstates. A concrete implementation in a recently proposed $mathcal{PT}$ symmetric waveguide system is suggested as an example of how to experimentally check the composition law and show the non-abelian nature of non-hermitian systems with multiple EPs.
We revisit the fundamental notion of continuity in representation theory, with special attention to the study of quantum physics. After studying the main theorem in the context of representation theory, we draw attention to the significant aspect of continuity in the analytic foundations of Wigner work. We conclude the paper by reviewing the connection between continuity, the possibility of defining certain local groups, and their relation to projective representations.
Let $mathcal{M}(n,m;F bp^n)$ be the configuration space of $m$-tuples of pairwise distinct points in $F bp^n$, that is, the quotient of the set of $m$-tuples of pairwise distinct points in $F bp^n$ with respect to the diagonal action of ${rm PU}(1,n;F)$ equipped with the quotient topology. It is an important problem in hyperbolic geometry to parameterize $mathcal{M}(n,m;F bp^n)$ and study the geometric and topological structures on the associated parameter space. In this paper, by mainly using the rotation-normalized and block-normalized algorithms, we construct the parameter spaces of both $mathcal{M}(n,m; bhq)$ and $mathcal{M}(n,m;bp(V_+))$, respectively.
The genesis of lasing, as an evolution of the laser hybrid eigenstates comprised of electromagnetic modes and atomic polarization, is considered. It is shown that the start of coherent generation at the laser threshold is preceded by the formation of a special hybrid state at the lasing pre-threshold. This special state is characterized by an enhanced coupling among excited atoms and electromagnetic modes. This leads to an increase in the rate of stimulated emission in the special state and, ultimately, to lasing. At the lasing pre-threshold, the transformation of hybrid eigenstates has the features of an exceptional point (EP) observed in non-Hermitian systems. The special state is formed when eigenfrequencies of two hybrid states coalesce or come close to each other. Below the pre-threshold, lifetimes of all hybrid states grow with increasing pump rate. When the pump rate crosses the pre-threshold, resonance trapping occurs with the lifetime of the special state continuing to increase while the lifetimes of all other eigenstates begin to decrease. Consequently, the latter eigenstates do not participate in the lasing. Thus, above the pre-threshold, a laser transitions into the single-mode regime.
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