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Register a new userDiscussion of the adiabatic hypothesis in control schemes using exceptional points
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We present calculations for the action of laser pulses on vibrational transfer within the H2+ and Na2 molecules in the presence of dissipation due to photodissociation of the molecule. The laser fields perform closed loops surrounding exceptional points in the laser parameter plane of intensity and wavelength. In principle the process should produce controlled vibrational transfers due to an adiabatic flip of the dressed eigenstates. We directly solve the Schrodinger equation with the complete time-dependent field instead of using the adiabatic Floquet formalism which initially suggested the design of the laser pulses. Results given by wavepacket propagations disagree with predictions obtained using the adiabatic hypothesis. Thus we show that there are large non-adiabatic exchanges and that the dissipative character of the dynamics renders the adiabatic flip very difficult to obtain. Using much longer durations than expected from previous studies, the adiabatic flip is only obtained for the Na2 molecule and with strong dissociation.
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The appearance of so-called exceptional points in the complex spectra of non-Hermitian systems is often associated with phenomena that contradict our physical intuition. One example of particular interest is the state-exchange process predicted for an adiabatic encircling of an exceptional point. In this work we analyse this and related processes for the generic system of two coupled oscillator modes with loss or gain. We identify a characteristic system evolution consisting of periods of quasi-stationarity interrupted by abrupt non-adiabatic transitions, and we present a qualitative and quantitative description of this switching behaviour by connecting the problem to the phenomenon of stability loss delay. This approach makes accurate predictions for the breakdown of the adiabatic theorem as well as the occurrence of chiral behavior observed previously in this context, and provides a general framework to model and understand quasi-adiabatic dynamical effects in non-Hermitian systems.
We present a formal geometric framework for the study of adiabatic quantum mechanics for arbitrary finite-dimensional non-degenerate Hamiltonians. This framework generalizes earlier holonomy interpretations of the geometric phase to non-cyclic states appearing for non-Hermitian Hamiltonians. We start with an investigation of the space of non-degenerate operators on a finite-dimensional state space. We then show how the energy bands of a Hamiltonian family form a covering space. Likewise, we show that the eigenrays form a bundle, a generalization of a principal bundle, which admits a natural connection yielding the (generalized) geometric phase. This bundle provides in addition a natural generalization of the quantum geometric tensor and derived tensors, and we show how it can incorporate the non-geometric dynamical phase as well. We finish by demonstrating how the bundle can be recast as a principal bundle, so that both the geometric phases and the permutations of eigenstates can be expressed simultaneously by means of standard holonomy theory.
An example of exceptional points in the continuous spectrum of a real, pseudo-Hermitian Hamiltonian of von Neumann-Wigner type is presented and discussed. Remarkably, these exceptional points are associated with a double pole in the normalization factor of the Jost eigenfunctions normalized to unit flux at infinity. At the exceptional points, the two unnormalized Jost eigenfunctions are no longer linearly independent but coalesce to give rise to two Jordan cycles of generalized bound state eigenfunctions embedded in the continuum and a Jordan block representation of the Hamiltonian. The regular scattering eigenfunction vanishes at the exceptional point and the irregular scattering eigenfunction has a double pole at that point. In consequence, the time evolution of the regular scattering eigenfunction is unitary, while the time evolution of the irregular scattering eigenfunction is pseudounitary. The scattering matrix is a regular analytical function of the wave number $k$ for all $k$ including the exceptional points.
The state of a quantum system may be steered towards a predesignated target state, employing a sequence of weak $textit{blind}$ measurements (where the detectors readouts are traced out). Here we analyze the steering of a two-level system using the interplay of a system Hamiltonian and weak measurements, and show that $textit{any}$ pure or mixed state can be targeted. We show that the optimization of such a steering protocol is underlain by the presence of Liouvillian exceptional points. More specifically, for high purity target states, optimal steering implies purely relaxational dynamics marked by a second-order exceptional point, while for low purity target states, it implies an oscillatory approach to the target state. The phase transition between these two regimes is characterized by a third-order exceptional point.
H$_2^+$ is an ideal candidate for a detailed study of strong field coherent control strategies inspired by basic mechanisms referring to some specific photodissociation resonances. Two of them are considered in this work, namely: Zero-width resonances (ZWR) on one hand, and coalescing pairs of resonances at exceptional points (EP) on the other hand. An adiabatic transport theory based on Floquet Hamiltonian formalism is developed within the challenging context of multiphoton dynamics involving nuclear continua. It is shown that a rigorous treatment is only possible for ZWRs, whereas adiabatic transport mediated by EPs is subjected to restrictions. Numerical maps of resonance widths and non-adiabatic couplings in the laser parameter plane help in optimally shaping control pulses. Full time-dependent wavepacket dynamics shows the possibility of selective, robust filtration and vibrational population transfers, within experimental feasibility criteria.
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