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arXiv:1208.0143 · Journal of Mathematical Physics

Geometric phases in quantum control disturbed by classical stochastic processes

David Viennot

Published Aug 1, 2012• 1 citation
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Abstract

We describe the geometric (Berry) phases arising when some quantum systems are driven by control classical parameters but also by outer classical stochastic processes (as, for example, classical noises). The total geometric phase is then divided into an usual geometric phase associated with the control parameters and a second geometric phase associated with the stochastic processes. The geometric structure in which these geometric phases take place is a composite bundle (and not an usual principal bundle), which is explicitly built in this paper. We explain why the composite bundle structure is the more natural framework to study this problem. Finally, we treat a very simple example of a two-level atom driven by a phase modulated laser field with a phase instability described by a Gaussian white noise. In particular, we compute the average geometric phase issued from the noise.

DOI
10.1063/1.4746696

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We describe the geometric (Berry) phases arising when some quantum systems are driven by control classical parameters but also by outer classical stochastic processes (as, for example, classical noises). The total geometric phase is then divided into an usual geometric phase associated with the control parameters and a second geometric phase associated with the stochastic processes. The geometric structure in which these geometric phases take place is a composite bundle (and not an usual principal bundle), which is explicitly built in this paper. We explain why the composite bundle structure is the more natural framework to study this problem. Finally, we treat a very simple example of a two-level atom driven by a phase modulated laser field with a phase instability described by a Gaussian white noise. In particular, we compute the average geometric phase issued from the noise.
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.
Due to the potential application in quantum information process, geometric phase of interacting system arouse many interests. Some physicists concentrate on the system in pure classical envi- ronment, while others study the system in pure quantized environment. So a natural question is asked: how about an interacting system in composite environments made up of both classical and quantized field. In this letter, we analyze a quantum system composed of two interacting spins, of which one is in classical magnetic field and the other is in quantized field. First, classical magnetic field driven Berry phases for the whole system and subsystem are studied. The effect of couplings between particles and photon on these phases are analyzed. In comparison with the dynamical quantized field, We find that even a static quantized field in its vacuum state can also have an effect on Berry phase. Second, quantized field driven Berry phases for the whole system and sub- system are formulated, including both one and two mode of this field. The vacuum induced effects are elaborated, moreover compared with the constant vacuum induced phases in former papers, the counterpart in this letter varies according to classical magnetic field, couplings and other pa- rameters. For the two mode quantized field, the rigorous relationship between the concurrence and Berry phase for subsystem are built up.
Based on a generic quantum open system model, we study the geometric nature of decoherence by defining a complex-valued geometric phase through stochastic pure states describing nonunitary, noncyclic, and nonadiabatic evolutions. The ensemble average of the complex geometric phases for the pure stochastic states yields a conventional geometric phase together with an amplitude factor. We show that the decoherence process described by the decaying amplitude can be a geometric quantity independent of the system's dynamics. It is a remarkable fact that the geometric phase of a quantum system can serve as an ideal realization of quantum gates due to its robustness against dynamical errors; however, in this paper we show that, for some open quantum systems, a desirable geometric phase may be accompanied by an unwanted robust geometric decoherence factor. Two exactly solvable models are studied to demonstrate that, while the decoherence is a purely dynamical effect for a dephasing two-level model, the decoherence in a dissipative two-level model can be a geometric process. Finally, we show that such a geometric decoherence effect may be eliminated by a nonperturbative control scheme.
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.