We study the adiabatic dynamics of degenerate quantum states induced by loop paths in a control parameter space. The latter correspond to noisy trajectories if the system is weakly coupled to environmental modes. On top of conventional dynamic dephasing, we find a universal non-Abelian geometric dephasing (NAGD) contribution and express it in terms of the non-Abelian Berry connection and curvature. We show that NAGD implies either decay or amplification of coherences as compared to the coherences when only dynamic dephasing is present. The full NAGD matrix structure can be probed through interference experiments. We outline such a detection scheme for modified Majorana braiding setups.
We study the adiabatic dynamics of degenerate quantum states induced by loop paths in a control parameter space. The latter correspond to noisy trajectories if the system is weakly coupled to environmental modes. On top of conventional dynamic dephasing, we find a universal non-Abelian geometric dephasing (NAGD) contribution and express it in terms of the non-Abelian Berry connection and curvature. We show that NAGD implies either decay or amplification of coherences as compared to the coherences when only dynamic dephasing is present. The full NAGD matrix structure can be probed through interference experiments. We outline such a detection scheme for modified Majorana braiding setups.
To what extent can coupling to a dissipative environment hinder the emergence of a geometric phase? And does dissipation undermine its possibly non-Abelian nature? Here, the authors show that non-Abelian geometric phase survives the presence of an environment, albeit modified. The dissipative coupling is manifest in the non-Abelian dynamics. Moreover, it gives rise to non-Abelian geometric dephasing, which may suppress or enhance coherent contributions, depending on the order of adiabatic operations.
Non-Abelian braiding operations of quantum states have attracted substantial attention due to their great potentials for realizing topological quantum computations. The adiabatic version of quantum braiding is robust against systematic errors, yet will suffer from decoherence and dephasing effects due to a long evolution time. In this paper, we propose to realize the braiding process in a three-fold degenerate subspace of a seven-level system, where the non-Abelian effect can be detected by changing the orders of the braiding. We accelerate the adiabatic control through adding auxiliary coupling terms according to a shortcut to adiabatic theory for the non-Abelian case. Furthermore, by generalizing the parallel adiabatic passages, adiabatic control can be accelerated through only reshaping the original control waveforms and the effective pulses area will be significantly reduced. Therefore, the proposed schemes may provide an experimentally feasible way to investigate the non-Abelian braiding in atomic systems and the waveguide systems.
We develop a theory of adiabatic response for open systems gov-erned by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for resid-ual dissipation even when the ground state is protected by a spectral gap. We give quantum response a geometric interpretation in terms of Hilbert space projections: For a two level system and, more generally, for systems with suitable functional form of the dephasing, the dissi-pative and non-dissipative parts of the response are linked to a metric and to a symplectic form. The metric is the Fubini-Study metric and the symplectic form is the adiabatic curvature. When the metric and symplectic structures are compatible the non-dissipative part of the inverse matrix of response coefficients turns out to be immune to de-phasing. We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: The qubit, coherent states and a model of the integer quantum Hall effect. Two frameworks that provide insight and understanding of transport coef-ficients are Kubo’s theory of linear response [1, 2] and the theory of adiabatic response [3, 4, 5]. Both have a quantum version and a classical version, agree when there is overlap, and endow the non-dissipative transport coefficients with the geometric meaning of the adiabatic (Berry’s) curvature [6]. A notable success of this approach to “geometrization of quantum re-sponse ” has been its application to the integer quantum Hall effect where 1 ar
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.
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.
Concepts from non-Hermitian quantum mechanics have proven useful in understanding and manipulating a variety of classical systems, such as those encountered in optics, classical mechanics, and metamaterial design. Recently, the non-Hermitian analog of the Berry phase for adiabatic processes was experimentally measured. In non-Hermitian systems, the Berry phase can have an imaginary part, which contributes to the amplification or decay of the total wave intensity. When the imaginary part of the Berry curvature is zero, this geometric amplification factor is determined solely by the initial and final points of the adiabatic path in parameter space, and it does not depend on how these points are connected by the path. We list classes of non-Hermitian Hamiltonians where this path independence is guaranteed by suitable symmetries, and we find that, for some of these classes, the amplification factor can be written only in terms of the Petermann factors of the initial and final points. Our result can, in turn, be used to experimentally obtain the Petermann factor by observing how the norm of the wave function changes under adiabatic processes. We validate our theory using a couple of concrete examples of physical relevance.