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arXiv:quant-ph/0306178 · Physical Review Letters

Spin- 1/2 Geometric Phase Driven by Decohering Quantum Fields

Angelo C. M. Carollo, Ivette Fuentes, Marcelo França Santos, Vlatko Vedral

Published Jan 15, 2004• 100 citations
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Abstract

We calculate the geometric phase of a spin-1/2 system driven by one and two mode quantum fields subject to decoherence. Using the quantum jump approach, we show that the corrections to the phase in the no-jump trajectory are different when considering adiabatic and nonadiabatic evolutions. We discuss the implications of our results from both fundamental as well as quantum computational perspectives.

DOI
10.1103/physrevlett.92.020402

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We calculate the geometric phase of a spin-1/2 system driven by one and two mode quantum fields subject to decoherence. Using the quantum jump approach, we show that the corrections to the phase in the no-jump trajectory are different when considering adiabatic and nonadiabatic evolutions. We discuss the implications of our results from both fundamental as well as quantum computational perspectives.
Quantum computer takes advantages from properties of superposition and entanglement, which are also very fragile and easy to be error. But they may be destroyed easily by a process called decoherence. Therefore, how to overcome the effect of decoherence is key to implement quantum computer. Using the no-jump trajectory, Berry's and AA's formula for geometric phase, we calculate the geometric phase of a spin-1/2 system driven by one and two mode quantum fields subject to decoherence in the adiabatic and nonadiabatic case, respectively. By quantum jump method, we find that the corrections to its phase for the no-jump trajectory are different when considering adiabatic and nonadiabatic evolutions. Finally, we discuss the implications of our results from fundamental and quantum computational points.
We investigate the geometric phase for a quantum field driven by an external source and subjected to spontaneous decay from coupling to the reservoir. Starting from a coherent state, we analyze the effect of decoherence when the system undergoes a nonadiabatic cyclic evolution in phase space, and show that the lowest correction to the conventional geometric phase is quadratic in the decaying rate of the quantum field. This is in distinct contrast with the nonadiabatic geometric phase associated with the evolution of a spin system subjected to spontaneous decay, which contains the first-order correction term. We further show that the unconventional geometric phase is also robust to field decay.
We investigate the behaviour of the geometric phase of a bipartite system with intersubsystem coupling via a quantum jump approach, where one of the subsystems is driven by a one- or two-mode quantized field which is also subjected to decoherence. The results show that the first correction due to the decohering field is only quadratic in decaying rate λ when the evolution of the system is adiabatic, and the lowest order correction is related to the first order in λ when the evolution of the system is non-adiabatic. Moreover, an interesting interpretation of this phenomenon is given.
The quantum jump method for the calculation of geometric phase is reviewed. This is an operational method to associate a geometric phase to the evolution of a quantum system subjected to decoherence in an open system. The method is general and can be applied to many different physical systems, within the Markovian approximation. As examples, two main source of decoherence are considered: dephasing and spontaneous decay. It is shown that the geometric phase is to very large extent insensitive to the former, i.e. it is independent of the number of jumps determined by the dephasing operator.