Berry’s geometrical phases in ESR in the presence of a stochastic process

Gamliel, Dan; Freed, Jack H.

doi:10.1103/physreva.39.3238

Cited by 59 publications

(44 citation statements)

References 42 publications

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“…For this reason, it is important to understand the interplay of adiabaticity and decoherence, which was studied in several publications in connection to Berry's phase [5][6][7][8][9]. In these, the authors usually postulated a Markovian master equation of Lindblad form.…”

Section: Introductionmentioning

confidence: 99%

Time-dependent Markovian master equation for adiabatic systems and its application to Cooper-pair pumping

Kamleitner¹,

Shnirman²

2011

Phys. Rev. B

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For adiabatically and periodically manipulated dissipative quantum systems we derive, using Floquet theory, a simple Markovian master equation. Contrary to some previous works we explicitly take into account the time dependence of the Hamiltonian and, therefore, obtain a master equation with a time-dependent dissipative part. We illustrate our theory with two examples and compare our results with the previously proposed master equations. In particular, we consider the problem of Cooper pair pumping and demonstrate the inadequacy of the secular (rotating wave) approximation when calculating the pumped charge. The secular approximation producing a master equation of the Lindblad type approximates well the quantum state (density matrix) of the system, while to determine the pumped charge a non-Lindblad master equation beyond the rotating wave approximation is necessary.

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Section: Introductionmentioning

confidence: 99%

Time-dependent Markovian master equation for adiabatic systems and its application to Cooper-pair pumping

Kamleitner¹,

Shnirman²

2011

Phys. Rev. B

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“…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%

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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“…The first complete open systems analyses of geometric phase for a mixed state, from two different perspectives, is to be found in the papers of Ellinas et al [15] and Gamliel and Freed [16]. The former worked with the standard master equation for the density operator of a multilevel atom subject to radiative damping and driven by a laser field with a time-dependent phase.…”

Section: Introductionmentioning

confidence: 99%

“…The system Hamiltonian was allowed to vary adiabatically, with the result that a non-degenerate eigenmatrix acquires a geometric phase as well as a dynamic phase. In [16], the effect of the environment was modelled as an external classical stochastic influence which, when averaged, gives rise to the relaxation terms of the master equation for the system. In both cases the effects of any geometric phase was then shown to be present in measurable quantities such as the inversion of a two state system.…”

Section: Introductionmentioning

confidence: 99%

“…The approaches used involve either solving the master equation of the system [13,14,15,16], or employing a quantum trajectory analysis [22,23,24] to unravel the dynamics into pure state trajectories, and calculating the geometric phases associated with individual pure state trajectories. Noise of a classical origin, such as stochastic fluctuations of the parameters of the Hamiltonian have also been studied by Chiara and Palma [25].…”

Section: Introductionmentioning

confidence: 99%

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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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Berry’s geometrical phases in ESR in the presence of a stochastic process

Cited by 59 publications

References 42 publications

Time-dependent Markovian master equation for adiabatic systems and its application to Cooper-pair pumping

Time-dependent Markovian master equation for adiabatic systems and its application to Cooper-pair pumping

Geometric phase in dephasing systems

Geometric phase for an adiabatically evolving open quantum system

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