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arXiv:1209.5305 · Physical Review B

Topological phases in adiabatic and nonadiabatic driven systems

Álvaro Gómez-León, Gloria Platero

Published Sep 12, 2012• 27 citations
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Abstract

In this work we study the geometrical and topological properties of nonequilibrium quantum systems driven by ac fields. We consider two tunnel-coupled spin qubits driven by either spatially homogeneous or inhomogeneous ac fields. Our analysis is an extension of the classical model introduced by Berry with the addition of the spatial degree of freedom. We calculate the Berry and Aharonov-Anandan geometric phases, and demonstrate the influence of the different field parameters in the geometric properties. We also discuss the topological properties associated with the different driving regimes, and show that by tuning the different parameters one can induce topological phase transitions, even in the nonadiabatic regime.

DOI
10.1103/physrevb.86.115318

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In this work we study the geometrical and topological properties of nonequilibrium quantum systems driven by ac fields. We consider two tunnel-coupled spin qubits driven by either spatially homogeneous or inhomogeneous ac fields. Our analysis is an extension of the classical model introduced by Berry with the addition of the spatial degree of freedom. We calculate the Berry and Aharonov-Anandan geometric phases, and demonstrate the influence of the different field parameters in the geometric properties. We also discuss the topological properties associated with the different driving regimes, and show that by tuning the different parameters one can induce topological phase transitions, even in the nonadiabatic regime.
Quantum computer takes advantages from properties of superposition and entanglement, which are also very fragile and easy to be error. But they may be destroyed easily by a process called decoherence. Therefore, how to overcome the effect of decoherence is key to implement quantum computer. Using the no-jump trajectory, Berry's and AA's formula for geometric phase, we calculate the geometric phase of a spin-1/2 system driven by one and two mode quantum fields subject to decoherence in the adiabatic and nonadiabatic case, respectively. By quantum jump method, we find that the corrections to its phase for the no-jump trajectory are different when considering adiabatic and nonadiabatic evolutions. Finally, we discuss the implications of our results from fundamental and quantum computational points.
We show the existence of a nonadiabatic geometric phase, i.e., an Aharonov-Anandan (AA) phase, in the Aharonov-Casher (AC) topological interference effect in one-dimensional mesoscopic rings. We find the AC phase is the phase accumulated by the spin wave function during a cyclic evolution, and show it is the sum of a geometric AA phase and a dynamical phase. In the adiabatic limit, the AA phase becomes the spin-orbit Berry phase introduced by Aronov and Lyanda-Geller. By solving exactly the model of a quasi-one-dimensional ring formed by the 2DEG on a semiconductor heterostructure, we discuss the observability of the AA phase in the AC effect.
If the time evolution of a quantum state leads back to the initial state, a geometric phase is accumulated that is known as the Berry phase for adiabatic evolution or as the Aharonov-Anandan (AA) phase for nonadiabatic evolution. We evaluate these geometric phases using Floquet theory for systems in time-dependent external fields with a focus on paths leading through a degeneracy of the eigenenergies. Contrary to expectations, the low-frequency limits of the two phases do not always coincide. This happens as the degeneracy leads to a slow convergence of the quantum states to adiabaticity, resulting in a nonzero finite or divergent contribution to the AA phase. Steering the system adiabatically through a degeneracy provides control over the geometric phase as it can cause a π shift of the Berry phase. On the other hand, we revisit an example of degeneracy crossing proposed by AA. We find that, at suitable driving frequencies, both geometric-phase definitions give the same result and the dynamical phase is zero due to the symmetry of time evolution about the point of degeneracy, providing an advantageous setup for manipulation of quantum states.
We present new developments in nonadiabatic geometric phases along two lines for systems undergoing changes of quantum state in intense fields. We first present a geometric representation of the non-Hermitian Schrodinger equation and introduce the notion of a complex multiphoton Aharonov-Anandan (AA) phase associated with dissipative two-level systems driven by periodic fields. The concept is further extended to include field modulation effects. We then develop the AA phase for spin-j systems in periodic fields and find conditions for cyclic evolution for general multi-level systems. In both cases, generalizations of the Floquet formalism lead to general analytical expressions for geometric phases that can be tested by experiments.
We address the development of geometric phases in classical and quantum magnetic moments (spin-1/2) precessing in an external magnetic field. We show that nonadiabatic dynamics lead to a topological phase transition determined by a change in the driving field topology. The transition is associated with an effective geometric phase which is identified from the paths of the magnetic moments in a spherical geometry. The topological transition presents close similarities between SO(3) and SU(2) cases but features differences in, e.g. the adiabatic limits of the geometric phases, being 2π and π in the classical and the quantum case, respectively. We discuss possible experiments where the effective geometric phase would be observable.