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AIP conference proceedings

Berry phase in the presence of external noise

Robert S. Whitney

Published Jan 1, 2005• 0 citations
Publisher

Abstract

Here we summarise and place in context two of our earlier works. There we investigate the geometric phase or Berry phase (BP) acquired by a spin‐half which is subject to external noise in addition to a slowly varying magnetic field (which generates the Berry phase). The noise may be due to the fluctuations of either quantum or classical degrees‐of‐freedom in the vicinity of the spin. We find that the noise modifies the Berry phase and that this modification is of a geometric nature. While the original BP (for an isolated system) is the flux of a monopole‐field through the loop traversed by the magnetic field, the noise‐induced modification of the BP is the flux of a quadrupole‐like field. The noise‐induced phase is complex and its imaginary part giving a geometric contribution to dephasing; its sign depends on the direction of the loop. Unlike the BP, this geometric dephasing is gauge invariant for open paths of the magnetic field. We discuss the consequences of this for proposals to use Berry phases to control qubits, in particular solid state (superconducting) qubits.

DOI
10.1063/1.2138615

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Here we summarise and place in context two of our earlier works. There we investigate the geometric phase or Berry phase (BP) acquired by a spin‐half which is subject to external noise in addition to a slowly varying magnetic field (which generates the Berry phase). The noise may be due to the fluctuations of either quantum or classical degrees‐of‐freedom in the vicinity of the spin. We find that the noise modifies the Berry phase and that this modification is of a geometric nature. While the original BP (for an isolated system) is the flux of a monopole‐field through the loop traversed by the magnetic field, the noise‐induced modification of the BP is the flux of a quadrupole‐like field. The noise‐induced phase is complex and its imaginary part giving a geometric contribution to dephasing; its sign depends on the direction of the loop. Unlike the BP, this geometric dephasing is gauge invariant for open paths of the magnetic field. We discuss the consequences of this for proposals to use Berry phases to control qubits, in particular solid state (superconducting) qubits.
We investigate the geometric phase or Berry phase acquired by a spin half which is both subject to a slowly varying magnetic field and weakly coupled to a dissipative environment (either quantum or classical). We study how this phase is modified by the environment and find that the modification is of a geometric nature. While the original Berry phase (for an isolated system) is the flux of a monopole field through the loop traversed by the magnetic field, the environment-induced modification of the phase is the flux of a quadrupolelike field. We find that the environment-induced phase is complex, and its imaginary part is a geometric contribution to dephasing. Its sign depends on the direction of the loop. Unlike the Berry phase, this geometric dephasing is gauge invariant for open paths of the magnetic field.
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.
It has been shown that geometric phases may be generated in a quantum system subject to noise by adiabatic manipulations of the fluctuating fields, e.g., by variation of the system-environment coupling. For a two-state quantum system, this phase is expressed in terms of the geometry of the path, traversed by the slowly varying direction and amplitude of the fluctuations. The origin of this phase and the possibilities of separating it from the known environment-induced modification of the Berry phase are discussed.
The geometric phase has been proposed as a candidate for noise resilient coherent manipulation of fragile quantum systems. Since it is determined only by the path of the quantum state, the presence of noise fluctuations affects the geometric phase in a different way than the dynamical phase. We have experimentally tested the robustness of Berry's geometric phase for spin-1/2 particles in a cyclically varying magnetic field. Using trapped polarized ultracold neutrons, it is demonstrated that the geometric phase contributions to dephasing due to adiabatic field fluctuations vanish for long evolution times.
Berry's phase often appears in quantum two-level systems with a degeneracy. An example of such a system is a spin-1/2 particle in a magnetic field. As the magnetic field is slowly evolved through a closed path, the particle has been shown to acquire an additional phase called Berry's phase, in addition to the usual dynamical phase. This phase has been found in two-level quantum systems intrinsic to many superconducting qubits and has particularly been calculated for the charge, flux and Josephson phase qubit. Here, we present an alternative derivation of the Berry's phase in a current-biased Josephson junction qubit. We also calculate the complete Berry's phase evaluated over a closed loop in a frequency parameter space and compare this with the geometric phase. From this comparison, we examine the possibility of using a single phase qubit for topological quantum computing.
We present measurements of the Berry Phase in a single solid-state spin qubit associated with the nitrogen-vacancy center in diamond. Our results demonstrate the remarkable degree of coherent control achievable in the presence of a highly complex solid-state environment. We manipulate the spin qubit geometrically by careful application of microwave radiation that creates an effective rotating magnetic field, and observe the resulting phase via spin-echo interferometry. We find good agreement with Berry's predictions within experimental errors. We also investigated the role of the environment on the geometric phase, and observed that unlike other solid-state qubit systems, the dephasing was primarily dominated by fast radial fluctuations in the path.