Berry's phase in magnetic resonance

Using the Born-Oppenheimer approximation, we derive an effective Hamiltonian for an optomechanical system that leads to a nonlinear Kerr effect in the system's vacuum. The oscillating mirror at one edge of the optomechanical system induces a squeezing effect in the intensity spectrum of the cavity field. A near-resonant laser field is applied at the other edge to drive the cavity field, in order to enhance the Kerr effect. We also propose a quantum-nondemolition-measurement setup to monitor a system with two cavities separated by a common oscillating mirror, based on our effective Hamiltonian approach.

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“…, which is very similar to the NMR experiments used to demonstrate the Berry phase [17]. In the rotating frame of reference, the effec-…”

Section: Induced Kerr Nonlinearity and Born-oppenheimer Approximasupporting

confidence: 60%

Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system

et al. 2009

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“…[17] in adiabatic regime by subjecting a nuclear spin to an effective magnetic field which slowly sweeps a cone. A similar approach was adopted by Jones et.al.…”

mentioning

confidence: 99%

Use of non-adiabatic geometric phase for quantum computing by NMR

Das

Kumar

2005

Journal of Magnetic Resonance

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Geometric phases have stimulated researchers for its potential applications in many areas of science. One of them is fault-tolerant quantum computation. A preliminary requisite of quantum computation is the implementation of controlled logic gates by controlled dynamics of qubits. In controlled dynamics, one qubit undergoes coherent evolution and acquires appropriate phase, depending on the state of other qubits. If the evolution is geometric, then the phase acquired depend only on the geometry of the path executed, and is robust against certain types of errors. This phenomenon leads to an inherently fault-tolerant quantum computation. Here we suggest a technique of using non-adiabatic geometric phase for quantum computation, using selective excitation. In a two-qubit system, we selectively evolve a suitable subsystem where the control qubit is in state |1 , through a closed circuit. By this evolution, the target qubit gains a phase controlled by the state of the control qubit. Using these geometric phase gates we demonstrate implementation of Deutsch-Jozsa algorithm and Grover's search algorithm in a two-qubit system.

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“…Note also that apart from the Berry phase some higher order corrections to the Born -Oppenheimer approximation (traditionally slightly confusing referred as geometric magnetism or deterministic friction, see [52]) also occur. The practically useful application of the Berry phase conception is the energy level displacements predicted in [53] and observed by NMR [54]. The essential physics of the phenomena can be illustrated as follows.…”

Section: Born -Oppenheimer Approximationmentioning

confidence: 99%

Instanton versus traditional WKB approach to the Landau-Zener problem

Benderskiĭ¹,

Vetoshkin²,

Kats³

2003

J. Exp. Theor. Phys.

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Different theoretical approaches to the famous two state Landau -Zener problem are briefly discussed. Apart from traditional methods of the adiabatic perturbation theory, Born -Oppenheimer approximation with geometric phase effects, two-level approach, and momentum space representation, the problem is treated semiclassically also in the coordinate space. Within the framework of the instanton approach we present a full and unified description of 1D Landau-Zener problem of level crossing. The method enables us to treat accurately all four transition points (appearing at two levels crossing), while the standard WKB approach takes into account only two of them. The latter approximation is adequate for calculating of the transition probability or for studying of scattering processes, however it does not work for finding corresponding chemical reactions rates, where very often for typical range of parameters all four transition points can be relevant. Applications of the method and of the results may concern the various systems in physics, chemistry and biology.

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Berry's phase in magnetic resonance

Cited by 131 publications

References 19 publications

Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system

Effective Hamiltonian approach to the Kerr nonlinearity in an optomechanical system

Use of non-adiabatic geometric phase for quantum computing by NMR

Instanton versus traditional WKB approach to the Landau-Zener problem

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