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Register a new userEncircling an Exceptional Point
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Authors
C. Dembowski
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We calculate analytically the geometric phases that the eigenvectors of a parametric dissipative two-state system described by a complex symmetric Hamiltonian pick up when an exceptional point (EP) is encircled. An EP is a parameter setting where the two eigenvalues and the corresponding eigenvectors of the Hamiltonian coalesce. We show that it can be encircled on a path along which the eigenvectors remain approximately real and discuss a microwave cavity experiment, where such an encircling of an EP was realized. Since the wavefunctions remain approximately real, they could be reconstructed from the nodal lines of the recorded spatial intensity distributions of the electric fields inside the resonator. We measured the geometric phases that occur when an EP is encircled four times and thus confirmed that for our system an EP is a branch point of fourth order.
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Dynamically varying system parameters along a path enclosing an exceptional point is known to lead to chiral mode conversion. But is it necessary to include this non-Hermitian degeneracy inside the contour for this process to take place? We show that a slow enough variation of parameters, even away from the systems exceptional point, can also lead to a robust asymmetric state exchange. To study this process, we consider a prototypical two-level non-Hermitian Hamiltonian with a constant coupling between elements. Closed form solutions are obtained when the amplification/attenuation coefficients in this arrangement are varied in conjunction with the resonance detuning along a circular contour. Using asymptotic expansions, this input-independent mode conversion is theoretically proven to take place irrespective of whether the exceptional point is enclosed or not upon encirclement. Our results significantly broaden the range of parameter space required for the experimental realization of such chiral mode conversion processes.
We show that a two-level non-Hermitian Hamiltonian with constant off-diagonal exchange elements can be analyzed exactly when the underlying exceptional point is perfectly encircled in the complex plane. The state evolution of this system is explicitly obtained in terms of an ensuing transfer matrix, even for large encirclements, regardless of adiabatic conditions. Our results clearly explain the direction-dependent nature of this process and why in the adiabatic limit its outcome is dominated by a specific eigenstate irrespective of initial conditions. Moreover, numerical simulations suggest that this mechanism can still persist in the presence of nonlinear effects. We further show that this robust process can be harnessed to realize an optical omni-polarizer: a configuration that generates a desired polarization output regardless of the input polarization state, while from the opposite direction it always produces the counterpart eigenstate.
Exceptional points (EPs) associated with a square-root singularity have been found in many non-Hermitian systems. In most of the studies, the EPs found are isotropic meaning that the same singular behavior is obtained independent of the direction from which they are approached in the parameter space. In this work, we demonstrate both theoretically and experimentally the existence of an anisotropic EP in an acoustic system that shows different singular behaviors when the anisotropic EP is approached from different directions in the parameter space. Such an anisotropic EP arises from the coalescence of two square-root EPs having the same chirality.
We consider a two-dimensional nonlinear waveguide with distributed gain and losses. The optical potential describing the system consists of an unperturbed complex potential depending only on one transverse coordinate, i.e., corresponding to a planar waveguide, and a small non-separable perturbation depending on both transverse coordinates. It is assumed that the spectrum of the unperturbed planar waveguide features an exceptional point (EP), while the perturbation drives the system into the unbroken phase. Slightly below the EP, the waveguide sustains two-component envelope solitons. We derive one-dimensional equations for the slowly varying envelopes of the components and show their stable propagation. When both traverse directions are taken into account within the framework of the original model, the obtained two-component bright solitons become metastable and persist over remarkably long propagation distances.
Recently, sensors with resonances at exceptional points (EPs) have been suggested to have a vastly improved sensitivity due to the extraordinary scaling of the complex frequency splitting of the $n$ initially degenerate modes with the $n$-th root of the perturbation. We show here that the resulting quantum-limited signal to noise at EPs is proportional to the perturbation, and comparable to other sensors, thus providing the same precision. The complex frequency splitting close to EPs is therefore not suited to estimate the precision of EP sensors. The underlying reason of this counter-intuitive result is that the mode fields, described by the eigenvectors, are equal for all modes at the EP, and are strongly changing with the perturbation.
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