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arXiv:quant-ph/0305054 · Physical Review Letters

Observation of Geometric Phases for Mixed States using NMR Interferometry

Jiangfeng Du, Ping Zou, Mingjun Shi, L. C. Kwek, Jian-Wei Pan, C. H. Oh, ARTUR K. EKERT, Daniel K. L. Oi, Marie Ericsson

Published Sep 5, 2003• 157 citations

Abstract

Examples of geometric phases abound in many areas of physics. They offer both fundamental insights into many physical phenomena and lead to interesting practical implementations. One of them, as indicated recently, might be an inherently fault-tolerant quantum computation. This, however, requires one to deal with geometric phases in the presence of noise and interactions between different physical subsystems. Despite the wealth of literature on the subject of geometric phases very little is known about this very important case. Here we report the first experimental study of geometric phases for mixed quantum states. We show how different they are from the well-understood, noiseless, pure-state case.

DOI: 10.1103/physrevlett.91.100403

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Geometric phase may enable inherently fault-tolerant quantum computation. However, due to potential decoherence effects, it is important to understand how such phases arise for mixed input states. We report the first experiment to measure mixed-state geometric phases in optics, using a Mach-Zehnder interferometer, and polarization mixed states that are produced in two different ways: decohering pure states with birefringent elements; and producing a nonmaximally entangled state of two photons and tracing over one of them, a form of remote state preparation.

Geometrical phases have been applied in virtually every major branch of physics and they play an important role in topology and knot theory in mathematics and quantum computation. However, most of the early works focus on pure quantum states which are very unrealistic and almost never occur in practice. The two existed definitions -Uhlmann's and Sjöqvist's- would result in different values for the geometric phase in general. The definition of geometric phase in mixed state scenario is still an open question. Here we present a unified framework for both approaches within one and the same formalism based on simple interferometry. We then present experimental results which confirm both approaches to mixed state geometric phase and clearly demonstrate that their unification is possible. Our experiments are furthermore the first such to measure Uhlmann's mixed state geometric phase and show in addition that it is different to the Sjöqvist's phase.

Experimental detection of geometric phase effects is based on interferometry techniques. The observation of a nonadiabatic geometric phase in nuclear magnetic resonance has been reported by D. Suter, K. T. Mueller, and A. Pines (1988, Phys. Rev. Lett., 60, 1218). A rigorous interpretation of their results is presented here. The experiment exploits the concept of fictitious spin-1/2 subsystems embedded in a spin system of higher order. A relevant expression for the geometric phase is derived here using the Euler rotation representation for the SU(2) group. In order to correlate the experimental data to the geometric phase, a decomposition of the density matrix in a basis of pure states is used and the isomorphism between the cyclic evolution of pure states and mixed states is constructed. Rigorous treatment of the dynamic phase, along with technical aspects of the experiment, shows that the observed echo shift is related directly to the geometric phase, in agreement with Suter et al. The observed geometric phase is classified as a frame-related phase.

We calculate the geometric phase for an open system (spin-boson model) which interacts with an environment (ohmic or nonohmic) at arbitrary temperature. However there have been many assumptions about the time scale at which the geometric phase can be measured, there has been no reported observation of geometric phases for mixed states under nonunitary evolution yet. We study not only how they are corrected by the presence of the different type of environments but estimate the corresponding times at which decoherence becomes effective as well. These estimations should be taken into account when planning experimental setups to study the geometric phase in the nonunitary regime, particularly important for the application of fault-tolerant quantum computation.