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Chinese Physics Letters

Effect of Noise on Berry's Phase for Quantum Computing

Jing Hui, Mo‐Lin Ge

Published May 16, 2003• 0 citations
Publisher

Abstract

Based on a new type of two-level-system reservoir, we investigate the impact of dissipation on Berry's phase of a spin trapped in a periodical external field. It is found that the existence of environmental noise could lead to a decaying term in the matrix of Berry's phase as the sign of a decoherence process, which is in agreement with the result of Nazir et al. (Phys. Rev. A 65 (2002) 042303). Particularly, in comparison with a specialized case of the traditional Leggett dissipation model, we only shows the dependence of time but not temperature in decaying term. A concrete case is exhibited by using the one-dimensional Ohmic function.

DOI
10.1088/0256-307x/20/6/302

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Based on a new type of two-level-system reservoir, we investigate the impact of dissipation on Berry's phase of a spin trapped in a periodical external field. It is found that the existence of environmental noise could lead to a decaying term in the matrix of Berry's phase as the sign of a decoherence process, which is in agreement with the result of Nazir et al. (Phys. Rev. A 65 (2002) 042303). Particularly, in comparison with a specialized case of the traditional Leggett dissipation model, we only shows the dependence of time but not temperature in decaying term. A concrete case is exhibited by using the one-dimensional Ohmic function.
Based on a new type of two-level-system reservoir, we investigate the impact of dissipation on Berry''s phase of a spin trapped in a periodical external field. It is found that the existence of environmental noise could lead to a decaying term in the matrix of Berry's phase as the sign of a decoherence process, which is in agreement with the result of Nazir et al. (Phys. Rev. A 65 (2002) 042303). Particularly, in comparison with a specialized case of the traditional Leggett dissipation model, we only shows the dependence of time but not temperature in decaying tem. A concrete case is exhibited by using the one-dimensional Ohmic function.
The geometric (Berry) phase of a two-level system in a dissipative environment is analyzed by using the second-quantized formulation, which provides a unified and gauge-invariant treatment of adiabatic and nonadiabatic phases and is thus applicable to a quantitative analysis of transitional regions away from ideal adiabaticity. In view of the recent experimental observation of the Berry phase in a superconducting qubit, we illustrate our formulation for a concrete adiabatic case in the Ohmic dissipation. The correction to the total phase together with the geometry-dependent dephasing time is given in a transparent way. The behavior of the geometric phase away from ideal adiabaticity is also analyzed in some detail.
We experimentally investigate the effects of noise on the adiabatic and cyclic geometric phase, also termed the Berry phase. By introducing artificial fluctuations in the path of the control field, we measure the geometric contribution to dephasing of an effective two-level system for a variety of noise powers and different paths. Our results, measured using a microwave-driven superconducting qubit, clearly show that only fluctuations which distort the path lead to geometric dephasing. In a direct comparison with the dynamic phase, which is path independent, we observe that the Berry phase is less affected by noise-induced dephasing. This observation directly points towards the potential of geometric phases for quantum gates or metrological applications.
We discuss the concept of the Berry phase in a dissipative system. We show that one can identify a Berry phase in a weakly-dissipative system and find the respective correction to this quantity, induced by the environment. This correction is expressed in terms of the symmetrized noise power and is therefore insensitive to the nature of the noise representing the environment, namely whether it is classical or quantum mechanical. It is only the spectrum of the noise which counts. We analyze a model of a spin-half (qubit) anisotropically coupled to its environment and explicitly show the coincidence between the effect of a quantum environment and a classical one.
We study the dynamics of the quantum phase distribution associated with the reduced density matrix of a system for a number of situations of practical importance, as the system evolves under the influence of its environment, interacting via a quantum nondemolition type of coupling, such that there is decoherence without dissipation, as well as when it interacts via a dissipative interaction, resulting in decoherence as well as dissipation. The system is taken to be either a two-level atom (or, equivalently, a spin-1∕21∕21∕2 system) or a harmonic oscillator, and the environment is modeled as a bath of harmonic oscillators, starting out in a squeezed thermal state. The impact of the different environmental parameters on the dynamics of the quantum phase distribution for the system starting out in various initial states is explicitly brought out. An interesting feature that emerges from our work is that the relationship between squeezing and temperature effects depends on the type of system-bath interaction. In the case of a quantum nondemolition type of interaction, squeezing and temperature work in tandem, producing a diffusive effect on the phase distribution. In contrast, in the case of a dissipative interaction, the influence of temperature can be counteracted by squeezing, which manifests as a resistance to randomization of phase. We make use of the phase distributions to bring out a notion of complementarity in atomic systems. We also study the variance of the phase using phase distributions conditioned on particular initial states of the system.