Complex geometrical phases for dissipative systems

Garrison, J. C.; Wright, E. M.

doi:10.1016/0375-9601(88)90905-x

Cited by 245 publications

(332 citation statements)

References 6 publications

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“…Garrison and Wright [8] were the first to touch on this issue by describing open system evolution in terms of a non-Hermitian Hamiltonian. This is a pure state analysis, so it did not address the problem of geometric phases for mixed states.…”

mentioning

confidence: 99%

“…Toward the geometric phase for mixed states in open systems, the approaches used involve solving the master equation of the system [9,10,11,12,13], employing a quantum trajectory analysis [14,15] or Krauss operators [16], and the perturbative expansions [17,18]. Some interesting results were achieved, briefly summarized as follows: nonhermitian Hamiltonian lead to a modification of Berry's phase [8,17], stochastically evolving magnetic fields produce both energy shift and broadening [18], phenomenological weakly dissipative Liouvillians alter Berry's phase by introducing an imaginary correction [11] or lead to damping and mixing of the density matrix elements [12]. However, almost all these studies are performed for dissipative systems, and thus the representations are applicable for systems whose energy is not conserved.…”

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confidence: 99%

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Geometric phase in dephasing systems

Wang

2005

Phys. Rev. A

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Beyond the quantum Markov approximation, we calculate the geometric phase of a two-level system driven by a quantized magnetic field subject to phase dephasing. The phase reduces to the standard geometric phase in the weak coupling limit and it involves the phase information of the environment in general. In contrast with the geometric phase in dissipative systems, the geometric phase acquired by the system can be observed on a long time scale. We also show that with the system decohering to its pointer states, the geometric phase factor tends to a sum over the phase factors pertaining to the pointer states.PACS numbers: 03.65. Vf, 03.65.Yz Quantum mechanics tell us that physical states are equivalent up to a global phase, which in general does not contain useful information about the described system and thus can be ignored. This is not the case, however, for a system transported round a circuit by varying the parameters s = (s 1 , s 2 , ...) in its Hamiltonian H( s). As Berry showed [1], the phase can have a component of geometric origin called geometric phase with important observable consequences, such as the Aharonov-Bohm effect [2] and the spin-1 2 particle driven by a rotating magnetic field [1]. The geometric phases that only depend on the path followed by the system during its evolution, have been investigated and tested in a variety of settings and have been generalized in several directions [3]. The geometric phases are attractive both from a theoretical perspective, and from the point of view of possible applications, among which geometric quantum computation [4,5,6,7] is one of the most importance.As realistic systems always interact with their environment, the study on the geometric phase in open systems become interesting. Garrison and Wright [8] were the first to touch on this issue by describing open system evolution in terms of a non-Hermitian Hamiltonian. This is a pure state analysis, so it did not address the problem of geometric phases for mixed states. Toward the geometric phase for mixed states in open systems, the approaches used involve solving the master equation of the system [9,10,11,12,13], employing a quantum trajectory analysis [14,15] or Krauss operators [16], and the perturbative expansions [17,18]. Some interesting results were achieved, briefly summarized as follows: nonhermitian Hamiltonian lead to a modification of Berry's phase [8,17], stochastically evolving magnetic fields produce both energy shift and broadening [18], phenomenological weakly dissipative Liouvillians alter Berry's phase by introducing an imaginary correction [11] or lead to damping and mixing of the density matrix elements [12]. However, almost all these studies are performed for dissipative systems, and thus the representations are applicable for systems whose energy is not conserved. For open systems with conserved energy (dephasing systems), the problem beyond the Markov approximation remains untouched to our best knowledge. Because the systemenvironment interaction H I and the free system Hamilton...

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confidence: 99%

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confidence: 99%

Geometric phase in dephasing systems

Wang

2005

Phys. Rev. A

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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: 85%

“…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%

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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“…Garrison and Wright [12] were the first to touch on this issue in a phenomenological way, by describing open system evolution in terms of a non-Hermitean Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Geometric phase for an adiabatically evolving open quantum system

2004

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We derive an elegant solution for a two-level system evolving adiabatically under the influence of a driving field with a time-dependent phase, which includes open system effects such as dephasing and spontaneous emission. This solution, which is obtained by working in the representation corresponding to the eigenstates of the time-dependent Hermitian Hamiltonian, enables the dynamic and geometric phases of the evolving density matrix to be separated and relatively easily calculated.

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Complex geometrical phases for dissipative systems

Cited by 245 publications

References 6 publications

Geometric phase in dephasing systems

Geometric phase in dephasing systems

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

Geometric phase for an adiabatically evolving open quantum system

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