Time-asymmetric quantum-state-exchange mechanism

Abstract

We show here that due to non-adiabatic couplings in decaying systems applying the same time-dependent protocol in the forward and reverse direction to the same mixed initial state leads to different final pure states. In particular, in laser driven molecular systems applying a specifically chosen positively chirped laser pulse or an equivalent negatively chirped laser pulse yields entirely different final vibrational states. This phenomenon occurs when the laser frequency and intensity are slowly varied around an exceptional point (EP) in the laser intensity and frequency parameter space where the non-hermitian spectrum of the problem is degenerate. The protocol implies that a positively chirped laser pulse traces a counter-clockwise loop in time in the laser parameters' space whereas a negatively chirped pulse follows the same loop in the clockwise direction. According to this protocol one can choose the final pure state from any initial state. The obtained results imply the intrinsic non-adiabaticity of quantum transport around an EP, and offer a way to observe the EP experimentally in time-dependent quantum systems.For open quantum systems where the effective Hamiltonian is non-hermitian (NH) the non crossing rule [1] is replaced by an intersection of two complex energy levels associated with two eigenfunctions of the NH Hamiltonian that have the same symmetry. Let us consider 2 × 2 Hamiltonian matrix H which depends on potential parameters q 1 and q 2 . These can be for instance the laser frequency and intensity when light interacts with two normal modes of a molecule. In open quantum system where the effective Hamiltonian is NH all matrix elements can attain complex values. The complex diagonal matrix elements are associated with metastable (resonance) states, such that −2ImH 11 and −2ImH 22 are the decay rates of the meta-stable states. The eigenvalues of this NH Hamiltonian are degenerate when ∆ = (H 11 − H 22 ) 2 + 4H 12 H 21 = 0 even though all matrix elements are different from zero. This situation is very different from the hermitian (standard) case where crossing requires H 12 = H 21 = 0 and H 11 = H 22 . At the crossing point a nonhermitian degeneracy (NHD) is obtained when the following two equations are satisfied:NHD is very different in its nature from hermitian degeneracy. NHD is obtained at the crossing point denoted by (q EP 1 , q EP 2 ), where the two eigenvalues coalesce and form a branch point (BP) in the complex energy spectrum. At the BP the first order derivatives of the eigenvalues with respect to q 1 or q 2 acquire infinitely large values (see for example Chapter 9 in Ref. [2]). This BP is also known as an exceptional point (EP) in the energy spectrum [3,4]. Moreover, at the BP (EP) not only the eigenvalues coalesce but also the corresponding eigenvectors. Such a phenomenon can never occur in standard QM. In NHQM as q 1 → q EP 1 and q 2 → q EP 2

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

“…This property of EP2s has already been demonstrated over a decade ago in a series of quasi-static microwave cavity experiments [54,[78][79][80][81] (see also Ref. [82]), although a true dynamical encirclement is a more complex problem [83,84] that has only been achieved in experiment very recently for the EP2B [85][86][87]. In App.…”

“…This much more involved calculation is carried out for the EP2B in Refs. [83,84], for example, in which it is demonstrated that the quasi-stationary method is insufficient to describe the dynamical evolution. However, we note that in Refs.…”

“…However, we note that in Refs. [83,84], the authors again rely on a heuristic 2x2 effective Hamiltonian. As we demonstrate in the main text of the present paper, that approach should be sufficient for the case of the EP2B (or the effectively mapped EP2A as in Refs.…”

“…(82) and (83). Combining the third line of this matrix equation with the normalization condition Ψ j |B|Ψ j = 1, we obtain the contribution of the d A site to the eigenket as…”

Model pseudofermionic systems: Connections with exceptional points

“…This hamiltonian was introduced quite recently in [8], and can be rewritten as 12) where Γ 1 and Γ 2 are positive quantities, ǫ 1 and ǫ 2 are reals, and ν 0 is complex-valued. It is a simple exercise to show that H GM M can be written as in (2.12) with the following identification: 13) where ∆ǫ = ǫ 2 − ǫ 1 , ∆Γ = Γ 2 − Γ 1 , ǫ = ǫ 2 + ǫ 1 and Γ = Γ 2 + Γ 1 .…”