Nonadiabatic transitions for a decaying two-level system: geometrical and dynamical contributions

Abstract

We study the Landau-Zener Problem for a decaying two-level-system described by a non-hermitean Hamiltonian, depending analytically on time. Use of a super-adiabatic basis allows to calculate the non-adiabatic transition probability P in the slow-sweep limit, without specifying the Hamiltonian explicitly. It is found that P consists of a "dynamical" and a "geometrical" factors. The former is determined by the complex adiabatic eigenvalues E±(t), only, whereas the latter solely requires the knowledge of α±(t), the ratio of the components of each of the adiabatic eigenstates. Both factors can be split into a universal one, depending only on the complex level crossing points, and a nonuniversal one, involving the full time dependence of E±(t). This general result is applied to the Akulin-Schleich model where the initial upper level is damped with damping constant γ. For analytic power-law sweeps we find that Stückelberg oscillations of P exist for γ smaller than a critical value γ c and disappear for γ > γ c . A physical interpretation of this behavior will be presented by use of a damped harmonic oscillator.

Adiabatic approximation for quantum dissipative systems: Formulation, topology, and superadiabatic tracking

“…This allows us to determine in principle the nonadiabatic coefficient c T (+∞). This generalizes the Dykhne-Davis-Pechukas (DDP) formula [20,21] to the dissipative case (see also [23] for a determination of the nonadiabatic coefficients for specific models using perturbation theory). 1 It is shown here that, due to the complex phase terms in (1), even in the case of a well-satisfied adiabatic condition, the two components of the solution can be of the same order.…”

“…We analyze adiabatic passage for this traceless Hamiltonian H using the formalism of the Stokes lines and the transition points (see also [23] for a derivation using perturbation theory).…”

“…and θ j is a geometrical phase [22,23,28]. In the case considered here, for which is real, θ j is 0 or π .…”

Non-Hermitian quantum annealing in the ferromagnetic Ising model

“…Here the critical point, z c , is determined as a solution of the equation, ε k (z c ) = 0, in the complex plane obtained by analytical continuation, t → z [30][31][32][33][34][35][36][37][38]. Similarly, if initially the system was in the excited state: |ψ k (0) = |u + (k, 0) , so that α k (0) = 0 and β k (0) = 1, the result of integration yields…”

Fresnel integrals and irreversible energy transfer in an oscillatory system with time-dependent parameters

“…Over last decades, growing attention has also been paid to non-Hermitian extensions of the classic theory, taking into account the effect of environment on two-level systems (e.g., [13][14][15] and references therein). Due to its generality, the Landau-Zener scenario has been applied to numerous problems in various contexts, such as laser physics [16], semiconductor superlattices [17], tunneling of optical [18] or acoustic [19,20] waves, and quantum information processing [21], to name just a few examples.…”