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arXiv:1302.3305 · Physical Review A

Exploring the effect of noise on the Berry phase

S. Berger, Marek Pechal, A. A. Abdumalikov, Christopher Eichler, L. Steffen, Arkady Fedorov, Andreas J. Wallraff, Stefan Filipp

Published Jun 26, 2013• 101 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.87.060303

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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.
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.
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.
While it is often thought that the geometric phase is less sensitive to fluctuations in the control fields, a very general feature of adiabatic Hamiltonians is the unavoidable dynamic phase that accompanies the geometric phase. The effect of control field noise during adiabatic geometric quantum gate operations has not been probed experimentally, especially in the canonical spin qubit system that is of interest for quantum information. We present measurement of the Berry phase and carry out adiabatic geometric phase gate in a single solid-state spin qubit associated with the nitrogen-vacancy center in diamond. We manipulate the spin qubit geometrically by careful application of microwave radiation that creates an effective rotating magnetic field, and observe the resulting Berry phase signal via spin echo interferometry. Our results show that control field noise at frequencies higher than the spin echo clock frequency causes decay of the quantum phase, and degrades the fidelity of the geometric phase gate to the classical threshold after a few (∼10) operations. This occurs inspite of the geometric nature of the state preparation, due to unavoidable dynamic contributions. We have carried out systematic analysis and numerical simulations to study the effects of the control field noise and imperfect driving waveforms on the quantum phase gate.
We introduce the nonadiabatic, or Aharonov-Anandan, geometric phase as a tool for quantum computation and show how this phase on one qubit can be monitored by a second qubit without any dynamical contribution. We also discuss how this geometric phase could be implemented with superconducting charge qubits. While the nonadiabatic geometric phase may circumvent many of the drawbacks related to the adiabatic (Berry) version of geometric gates, we show that the effect of fluctuations of the control parameters on nonadiabatic phase gates is more severe than for the standard dynamic gates. Similarly, fluctuations also affect to a greater extent quantum gates that use the Berry phase instead of the dynamic phase.
In quantum information science, the phase of a wave function plays an important role in encoding information. Although most experiments in this field rely on dynamic effects to manipulate this information, an alternative approach is to use geometric phase, which has been argued to have potential fault tolerance. We demonstrated the controlled accumulation of a geometric phase, Berry's phase, in a superconducting qubit; we manipulated the qubit geometrically by means of microwave radiation and observed the accumulated phase in an interference experiment. We found excellent agreement with Berry's predictions and also observed a geometry-dependent contribution to dephasing.
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.