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arXiv:0708.3891 · Journal of Physics A Mathematical and Theoretical

The brachistochrone problem in open quantum systems

I. Rotter

Published Nov 14, 2007• 15 citations
arXivHTMLPDF

Abstract

Recently, the quantum brachistochrone problem has been discussed in the literature by using non-Hermitian Hamilton operators of different types. Here, it is demonstrated that the passage time is tunable in realistic open quantum systems due to the biorthogonality of the eigenfunctions of the non-Hermitian Hamilton operator. As an example, the numerical results obtained by Bulgakov et al for the transmission through microwave cavities of different shapes are analyzed from the point of view of the brachistochrone problem. The passage time is shortened in the crossover from the weak-coupling to the strong-coupling regime where the resonance states overlap and many branch points (exceptional points) in the complex plane exist. The effect can not be described in the framework of the standard quantum mechanics with the Hermitian Hamilton operator and consideration of S matrix poles.

DOI
10.1088/1751-8113/40/48/014

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Recently the quantum brachistochrone problem has been considered for non-Hermitian PT\cal PTPT-symmetric quantum system, and it has been shown that the optimal evolution time required to transform a given initial state ∣ψi>|\psi_i>∣ψi​> into specific final state ∣ψf>|\psi_f>∣ψf​> can be made arbitrary small. Here we study the similar problem for generic non-Hermitian Hamiltonian and find that the same answer can be obtained. We show that at the exceptional point the distance between ∣ψi>|\psi_i>∣ψi​> and ∣ψf>|\psi_f>∣ψf​> becomes virtually vanishing and quantum tunneling occurring at the exceptional point yields the shortest timed required to evolve from the initial state ∣ψi>|\psi_i>∣ψi​> into the final state ∣ψf>|\psi_f>∣ψf​>. These results may have applications in holonomic quantum computation and quantum information processing.
Calculations for open quantum systems are performed usually by taking into account their embedding into one common environment, which is mostly the common continuum of scattering wave functions. Realistic quantum systems are coupled however often to more than one continuum. For example, the conductance of an open cavity needs at least two environments, namely the input and the output channel. In the present paper, we study generic features of an open quantum system coupled to two channels. We compare the results with those characteristic of a one-channel system. Of special interest is the parameter range which is influenced by singular (exceptional) points. Here, the states of the system are mixed via the environment and the dynamics is determined, in any case, by nonlinear processes. These processes restore the resonance structure of the cross section in the one-channel case, such that there is no need to consider them in calculations for concrete systems. In the two-channel case, however, they cause new effects. Most interesting is the enhanced conductance of an open cavity in this parameter range. This enhancement follows directly from the properties of the eigenstates of a non-Hermitian Hamilton operator (and is not related to the existence of a superradiant state). The study of the conductance allows, among others, an experimental test of the non-Hermitian quantum physics.
The PT\mathcal{P}\mathcal{T}PT-symmetric (PTS) quantum brachistochrone problem is re-analyzed as a quantum system consisting of a non-Hermitian PTS component and a purely Hermitian component simultaneously. Interpreting this specific setup as a subsystem of a larger Hermitian system, we find nonunitary operator equivalence classes (conjugacy classes) as natural ingredients which contain at least one Dirac-Hermitian representative. With the help of a geometric analysis the compatibility of the vanishing passage time solution of a PTS brachistochrone with the Anandan-Aharonov lower bound for passage times of Hermitian brachistochrones is demonstrated.
Non-Hermitian quantum physics is used successfully for the description of different puzzling experimental results, which are observed in open quantum systems. Mostly, the influence of exceptional points on the dynamical properties of the system is studied. At these points, two complex eigenvalues Ei≡Ei+iΓi/2 of the non-Hermitian Hamiltonian H coalesce (where Ei is the energy and Γi is the inverse lifetime of the state i ). We show that also the eigenfunctions Φi of the two states play an important role, sometimes even the dominant one. Besides exceptional points, other critical points exist in non-Hermitian quantum physics. At these points a=acr in the parameter space, the biorthogonal eigenfunctions of H become orthogonal. For illustration, we show characteristic numerical results.