Geometrical phase in the cyclic evolution of non-Hermitian systems

Dattoli, G.; Mignani, Roberto; Torre, A.

doi:10.1088/0305-4470/23/24/020

Cited by 94 publications

(81 citation statements)

References 50 publications

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“…In these models, the couplings between quantum systems and their neighborhoods in the "ab initio" Hamiltonians do not depend on the time variation of a set of classical parameters. This fact, togheter with result (35) (the condition to the existence of an imaginary geometric phase) put in check the correctness of the imaginary geometric phases in the literature due to dissipative effects [5,6,7]. In order to verify if the imaginary phase for a quantum model described by a phenomenological non-hermitian hamiltonian truly exists -being of true geometric origin, and not a fake one due to eqs.…”

Section: Discussionmentioning

confidence: 86%

“…The non-hermitian parts in the Hamiltonians in references [5,6,7] take into account the losses of a quantum system to its environment (a suitable reservoir of degrees of freedom at equilibrium). In these models, the couplings between quantum systems and their neighborhoods in the "ab initio" Hamiltonians do not depend on the time variation of a set of classical parameters.…”

Section: Discussionmentioning

confidence: 99%

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Adiabatic approximation in the density matrix approach: non-degenerate systems

Pinto¹,

Romero²,

Thomaz³

2002

Physica A: Statistical Mechanics and its Applications

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We study the adiabatic limit in the density matrix approach for a quantum system coupled to a weakly dissipative medium. The energy spectrum of the quantum model is supposed to be non-degenerate. In the absence of dissipation, the geometric phases for periodic Hamiltonians obtained previously by M.V. Berry are recovered in the present approach. We determine the necessary condition satisfied by the coefficients of the linear expansion of the non-unitary part of the Liouvillian in order to the imaginary phases acquired by the elements of the density matrix, due to dissipative effects, be geometric. The results derived are model-independent. We apply them to spin 1 2 model coupled to reservoir at thermodynamic equilibrium.

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Section: Discussionmentioning

confidence: 86%

Section: Discussionmentioning

confidence: 99%

Adiabatic approximation in the density matrix approach: non-degenerate systems

Pinto¹,

Romero²,

Thomaz³

2002

Physica A: Statistical Mechanics and its Applications

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“…These are not only outgoing solutions of the wave equation (since g(x, ω)e −iωt is such a solution for any ω), but satisfy the nodal condition at the origin as well (since f (x, ω)e −iωt has this property for any ω). The momenta associated with the f j,n arê 4) so that the action of the hamiltonian is H|f j,n = ω j |f j,n + |f j,n−1 , with |f j,−1 ≡ 0. For fixed j, the functions f j,n (x, t) (n = 0, .…”

Section: Jordan Blocksmentioning

confidence: 99%

“…Dissipative systems are often discussed, in a phenomenological way, by postulating a nonhermitian hamiltonian (NHH) H, whose left eigenvectors f j | and right eigenvectors |f j form a bi-orthogonal system (BS) [1,2,3,4,5]. It is usually assumed that these eigenvectors are complete; the BS then constitutes a bi-orthogonal basis (BB).…”

Section: Introduction a Dissipative Systems And Bi-orthogonal Basesmentioning

confidence: 99%

Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems

Brink¹,

Young

2001

J. Phys. A: Math. Gen.

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Dissipative systems can be described in terms of non-hermitian hamiltonians H, whose left eigenvectors f j | and right eigenvectors |f j form a bi-orthogonal system. Bi-orthogonal systems could suffer from two difficulties. (a) If the eigenvectors do not span the whole space, then H can only be diagonalized to blocks (the Jordan-block problem). (b) Normalization would not be possible and many familiar-looking formulas would fail if f j |f j = 0 for some j (the orthonormalization problem). Waves in open systems provide a well-founded realization of a bi-orthogonal system, and it is shown that these two problems can indeed occur and are both related to higher-order poles in the frequency-domain Green's function. The resolution is then given by introducing a generalized duality transformation involving extra basis vectors, whose time evolution is modified by polynomials in the time t. One thus obtains a nontrivial extension of the bi-orthogonal formalism for dissipative systems.

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“…This problem arises naturally in connection with various experiments which, by their very essence, require the observation of the geometric phase in metastable states. The Berry phase in the optical supermode propagation in a free laser, which is a classical system described by a Schrödinger-like equation with a non-Hermitian Hamiltonian, was studied by Dattoli et al [5]. The measurement of the geometric phase in atomic systems with two energy levels, one of which at least is metastable, was also described in terms of a non-Hermitian Hamiltonian by Miniatura et al [6].…”

Section: Introductionmentioning

confidence: 99%

Accidental degeneracy and berry phase of resonant states

Mondragón

Hernández

Irreversibility and Causality Semigroups and Rigged Hilbert Spaces

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We study the complex geometric phase acquired by the resonant states of an open quantum system which evolves irreversibly in a slowly time dependent environment. In analogy with the case of bound states, the Berry phase factors of resonant states are holonomy group elements of a complex line bundle with structure group C * . In sharp contrast with bound states, accidental degeneracies of resonances produce a continuous closed line of singularities formally equivalent to a continuous distribution of "magnetic" charge on a "diabolical" circle, in consequence, we find different classes of topologically inequivalent non-trivial closed paths in parameter space.

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Geometrical phase in the cyclic evolution of non-Hermitian systems

Cited by 94 publications

References 50 publications

Adiabatic approximation in the density matrix approach: non-degenerate systems

Adiabatic approximation in the density matrix approach: non-degenerate systems

Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems

Accidental degeneracy and berry phase of resonant states

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