© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Physica Scripta

Generalized Demkov model: geometrical adiabatic phase factors

S F C O'Rourke

Published Apr 1, 1992• 5 citations

Abstract

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.

DOI: 10.1088/0031-8949/45/4/002

Related papers

Circuit-dependent adiabatic phase factors from phase integral theory

Peter V. Coveney, Derrick S. F. Crothers, J. H. Macek

The authors show that circuit-dependent adiabatic phase factors occur naturally in the phase integral theory of atomic collisions, being a physical manifestation of the Stokes phenomenon familiar in asymptotic analysis. This implies a generalisation of Berry's work (1984) on geometric phase factors for situations involving adiabatic parallel transport around closed circuits in the complex plane.

Journal of Physics B Atomic Molecular and Optical Physics•Apr 14, 1988•6 citations

Geometric phase around exceptional points

Alexei A. Mailybaev, Oleg N. Kirillov, Alexander P. Seyranian

A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly $\ensuremath{\pi}$ for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to $\ensuremath{\pi}$ for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.

arXiv:quant-ph/0501040•Physical Review A•Jul 20, 2005•181 citations

Geometric amplitude factors in adiabatic quantum transitions

Michael Berry

Abstract The exponentially small probability of transition between two quantum states, induced by the slow change over infinite time of an analytic hamiltonian Ĥ = H(δt). Ŝ ( where δ is a small adiabatic parameter and Ŝ is the Vector spin -½ operator), contains an additional factor exp{ᴦg} of purely geometric origin (that is, independent of δ and ħ). For ᴦg to be non-zero, Ĥ must be complex hermitian rather than real symmetric, and the hamiltonian curve H(ז) must not lie in a plane through the origin nor be a helix identical (up to rigid motion) with its time reverse. An expression is given for ᴦg, as an integral from the real t axis to the complex time of degeneracy of the two states. Explicit examples are given. The geometric effect could be observed in experiments with spinning particles.

Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences•Aug 8, 1990•119 citations

Adiabatic theorem and generalized geometrical phase in the case of pseudo-Hermitian systems

S Cheniti, Walid Koussa, A Medjber +1 more

Abstract A generalization of the adiabatic theorem for quantum systems governed by pseudo-Hermitian Hamiltonians and details of its demonstration are given. Introducing a modified time-dependent metric giving a precise description of the quantum unitary evolution where the obtained effective Hamiltonian is observable because it is mean value is real. We show that an eigenstate of a pseudo-Hermitian Hamiltonian slowly transported will acquire a real generalized geometrical phase factor which contains two contributions: the first one corresponds to the conventional Berry’s phase as expected and a new geometrical term that we call the metric geometrical phase. We will apply our results in the cases of the famous time dependent brachistochrone problem and a non-Hermitian displaced harmonic oscillator.

Journal of Physics A Mathematical and Theoretical•Aug 7, 2020•11 citations

Non-Hermitian adiabatic transport in spaces of exceptional points

Judith Hoeller, N. Read, Jack Harris

We consider the space of $n\ifmmode\times\else\texttimes\fi{}n$ non-Hermitian Hamiltonians ($n=2$, 3, ...) that are equivalent to a single $n\ifmmode\times\else\texttimes\fi{}n$ Jordan block. We focus on adiabatic transport around a closed path (i.e., a loop) within this space, in the limit as the time scale $T=1/\ensuremath{\varepsilon}$ taken to traverse the loop tends to infinity. We show that, for a certain class of loops and a choice of initial state, the state returns to itself and acquires a complex phase that is ${\ensuremath{\varepsilon}}^{\ensuremath{-}1}$ times an expansion in powers of ${\ensuremath{\varepsilon}}^{1/n}$. The exponential of the term of $n\mathrm{th}$ order (which is equivalent to the ``geometric'' or Berry phase modulo $2\ensuremath{\pi}$) is thus independent of $\ensuremath{\varepsilon}$ as $\ensuremath{\varepsilon}\ensuremath{\rightarrow}0$; it depends only on the homotopy class of the loop and is an integer power of ${e}^{2\ensuremath{\pi}i/n}$. One of the conditions under which these results hold is that the state being transported is, for all points on the loop, that of slowest decay.

arXiv:1809.07175•Physical Review A•Sep 16, 2020•39 citations

Quantal phase factors accompanying adiabatic changes

Michael Berry

Abstract A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Mar 8, 1984•9,250 citations

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.

A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly $\ensuremath{\pi}$ for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to $\ensuremath{\pi}$ for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.

Abstract The exponentially small probability of transition between two quantum states, induced by the slow change over infinite time of an analytic hamiltonian Ĥ = H(δt). Ŝ ( where δ is a small adiabatic parameter and Ŝ is the Vector spin -½ operator), contains an additional factor exp{ᴦg} of purely geometric origin (that is, independent of δ and ħ). For ᴦg to be non-zero, Ĥ must be complex hermitian rather than real symmetric, and the hamiltonian curve H(ז) must not lie in a plane through the origin nor be a helix identical (up to rigid motion) with its time reverse. An expression is given for ᴦg, as an integral from the real t axis to the complex time of degeneracy of the two states. Explicit examples are given. The geometric effect could be observed in experiments with spinning particles.

Abstract A generalization of the adiabatic theorem for quantum systems governed by pseudo-Hermitian Hamiltonians and details of its demonstration are given. Introducing a modified time-dependent metric giving a precise description of the quantum unitary evolution where the obtained effective Hamiltonian is observable because it is mean value is real. We show that an eigenstate of a pseudo-Hermitian Hamiltonian slowly transported will acquire a real generalized geometrical phase factor which contains two contributions: the first one corresponds to the conventional Berry’s phase as expected and a new geometrical term that we call the metric geometrical phase. We will apply our results in the cases of the famous time dependent brachistochrone problem and a non-Hermitian displaced harmonic oscillator.

We consider the space of $n\ifmmode\times\else\texttimes\fi{}n$ non-Hermitian Hamiltonians (

n=2

, 3, ...) that are equivalent to a single $n\ifmmode\times\else\texttimes\fi{}n$ Jordan block. We focus on adiabatic transport around a closed path (i.e., a loop) within this space, in the limit as the time scale $T=1/\ensuremath{\varepsilon}$ taken to traverse the loop tends to infinity. We show that, for a certain class of loops and a choice of initial state, the state returns to itself and acquires a complex phase that is ${\ensuremath{\varepsilon}}^{\ensuremath{-}1}$ times an expansion in powers of ${\ensuremath{\varepsilon}}^{1/n}$. The exponential of the term of

n\mathrm{th}

order (which is equivalent to the ``geometric'' or Berry phase modulo $2\ensuremath{\pi}$) is thus independent of $\ensuremath{\varepsilon}$ as $\ensuremath{\varepsilon}\ensuremath{\rightarrow}0$; it depends only on the homotopy class of the loop and is an integer power of ${e}^{2\ensuremath{\pi}i/n}$. One of the conditions under which these results hold is that the state being transported is, for all points on the loop, that of slowest decay.

Abstract A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.