Generalized Demkov model: Geometric adiabatic phase factors

Norio Nakagawa

Norio Nakagawa. Ames Laboratory

S F C O'Rourke

We consider a two-state model which extends the exponential time-dependent Demkov model of non-adiabatic coupling to complex energies and interactions. Using the method of steepest descent, exact and practicable asymptotic expansions of the Hankel functions H v j (1) (λ), H v j (2) (λ) are obtained when | v j | and | λ | are large and of the same order. The semiclassical scattering two channel matrix within the strong-coupling Demkov model and embracing non-adiabatic parameters is derived. Closed form expressions for the transition probabilities, which incorporate the effects of the classical turning point, are also derived. The matrix elements are shown to reduce correctly in known limiting cases. These include the case of exact symmetric resonance, the Landau-Zener and Callaway-Bartling model and the Stueckelberg noncrossing formula. In the case of a Hamiltonian matrix which is complex Hermitian on the real t axis, Berry has shown that the phenomenon of anholonomy arises as the physical manifestation of a non-integrable real geometric phase in adiabatic parallel transport round a closed circuit in real space. More recently Berry has reinterpreted this theory as arising from transport around a circuit in complex time. The geometric amplitude factor, which is simply the anholonomic phase corresponding to the connection on a complex fibre bundle, occurs naturally in the Demkov model which is an example of a closed system in parameter space driven by a complex Hermitian Hamiltonian. It is shown that the Berry phase phenomenon also involves contributions from both real and complex circuit adiabatic phases.

Fanyuy Fai, Lukong Cornelius Fai, Afungchui David +1 more

Abstract Geometric phases and holonomies offer a robust route to quantum control and simulation. We introduce a high-fidelity quantum simulation framework based on the analytically solvable Demkov-Kunike (DK) model, which extends the standard Landau-Zener paradigm by providing richer control over evolution paths. We demonstrate how tailored DK pulse sequences can simulate the braiding statistics of Abelian anyons through Stückelberg interferometry, with a protocol for exact dynamical phase cancellation. Generalizing to a three-level Λ-system, we engineer a non-Abelian Wilczek-Zee connection and construct a universal set of holonomic quantum gates. A detailed analysis of the evolution operator and leakage probability for the three-level DK model is provided, identifying optimal adiabaticity regimes for high fidelity. We propose an experimental implementation using trapped ions, including a rigorous derivation of the effective DK Hamiltonian in a decoherence-free subspace. Numerical simulations confirm the expected interference signatures, gate non-commutativity, and enhanced robustness of geometric gates under decoherence. This work establishes DK pulses as a versatile tool for simulating topological-like phenomena and engineering geometric quantum gates in controllable quantum systems.

Shen, J Q, Chen, P, Hong Mao

The present paper finds the complete set of exact solutions of the general time-dependent dynamical models for quantum decoherence, by making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation. Based on this, the general explicit expression for the decoherence factor is then obtained and the adiabatic classical limit of an illustrative example is discussed. The result (i.e., the adiabatic classical limit) obtained in this paper is consistent with what obtained by other authors, and futhermore we obtain the more general results concerning the time-dependent non-adiabatic quantum decoherence. It is shown that the invariant theory is appropriate for treating both the time-dependent quantum decoherence and the geometric phase factor.

Anling Wang, Fuping Liu, Zhao-Xian Yu +1 more