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New Journal of Physics

Experimental verification of the inertial theorem control protocols

Chang-Kang Hu, Roie Dann, Jin‐Ming Cui, Yun‐Feng Huang, Chuan‐Feng Li, Guang‐Can Guo, Alan C. Santos, Ronnie Kosloff

Published Sep 1, 2021• 4 citations
Publisher

Abstract

An experiment based on a trapped ytterbium ion validates the inertial theorem for the SU (2) algebra. The qubit is encoded within the hyperfine states of the atom and controlled by RF fields. The inertial theorem generates analytical solutions for non-adiabatically driven systems that are 'accelerated' slowly, bridging the gap between the sudden and adiabatic limits. These solutions are shown to be stable to small deviations, both experimentally and theoretically. By encoding a two-level system into hyperphine structure of a trapped ytterbium, we explore the high control over the system dynamics in order to validate range of applicability of the inertial theorem in our system. For large deviations from the inertial condition, the experimental results show that the phase remains accurate while the amplitude diverges, so the inertial theorem has good robustness in the phase estimate. As a result, we experimentally showed that the inertial solutions pave the way to rapid quantum control of closed, as well as open quantum systems.

DOI
10.1088/1367-2630/ac2710

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An experiment based on a trapped ytterbium ion validates the inertial theorem for the SU (2) algebra. The qubit is encoded within the hyperfine states of the atom and controlled by RF fields. The inertial theorem generates analytical solutions for non-adiabatically driven systems that are 'accelerated' slowly, bridging the gap between the sudden and adiabatic limits. These solutions are shown to be stable to small deviations, both experimentally and theoretically. By encoding a two-level system into hyperphine structure of a trapped ytterbium, we explore the high control over the system dynamics in order to validate range of applicability of the inertial theorem in our system. For large deviations from the inertial condition, the experimental results show that the phase remains accurate while the amplitude diverges, so the inertial theorem has good robustness in the phase estimate. As a result, we experimentally showed that the inertial solutions pave the way to rapid quantum control of closed, as well as open quantum systems.
An experiment based on a trapped Ytterbium ion validates the inertial theorem for the SU(2) algebra. The qubit is encoded within the hyperfine states of the atom and controlled by RF fields. The inertial theorem generates analytical solutions for non-adiabatically driven systems that are `accelerated' slowly, bridging the gap between the sudden and adiabatic limits. These solutions are shown to be stable to small deviations, both experimentally and theoretically. As a result, the inertial solutions pave the way to rapid quantum control of closed, as well as open quantum systems. For large deviations from the inertial condition, the amplitude diverges while the phase remains accurate.
An experimental verification of the inertial theorem is presented involving two hyperfine states of a trapped Ytterbium ion. The theorem generates an analytical solution for non-adiabaticlly driven systems `accelerated' slowly, bridging the gap between the sudden and adiabatic limits. These solutions have shown to be stable to small deviations, both experimentally and theoretically. As a result, the inertial solutions pave the way to rapid quantum control of closed as well as open quantum systems. For large deviations from the inertial condition, the phase of the SU(2)SU(2)SU(2) algebra solution remains accurate, involving inaccuracies dominated by an amplitude difference.
We present a theorem describing stable solutions for a driven quantum system.The theorem, coined inertial theorem, is applicable for fast driving, provided the acceleration rate is small.The theorem states that in the inertial limit eigenoperators of the propagator remain invariant throughout the dynamics, accumulating dynamical and geometric phases.The proof of the theorem utilizes the structure of Liouville space and a closed Lie algebra of operators.We demonstrate applications of the theorem by studying three explicit solutions of a harmonic oscillator, two-level and three-level system models.These examples demonstrate that the inertial solution is superior to that obtained with the adiabatic approximation.Inertial protocols can be combined to generate a family of solutions.The inertial theorem is then employed to extend the validity of the Markovian master equation to strongly driven open quantum systems.In addition, we explore the consequence of geometric phases associated with the driving parameters.
Abstract Validity conditions for the adiabatic approximation are useful tools to understand and predict the quantum dynamics. Remarkably, the resonance phenomenon in oscillating quantum systems has challenged the adiabatic theorem. In this scenario, inconsistencies in the application of quantitative adiabatic conditions have led to a sequence of new approaches for adiabaticity. Here, by adopting a different strategy, we introduce a validation mechanism for the adiabatic approximation by driving the quantum system to a non-inertial reference frame. More specifically, we begin by considering several relevant adiabatic approximation conditions previously derived and show that all of them fail by introducing a suitable oscillating Hamiltonian for a single quantum bit (qubit). Then, by evaluating the adiabatic condition in a rotated non-inertial frame, we show that all of these conditions, including the standard adiabatic condition, can correctly describe the adiabatic dynamics in the original frame, either far from resonance or at a resonant point. Moreover, we prove that this validation mechanism can be extended for general multi-particle quantum systems, establishing the conditions for the equivalence of the adiabatic behavior as described in inertial or non-inertial frames. In order to experimentally investigate our method, we consider a hyperfine qubit through a single trapped Ytterbium ion 171Yb+, where the ion hyperfine energy levels are used as degrees of freedom of a two-level system. By monitoring the quantum evolution, we explicitly show the consistency of the adiabatic conditions in the non-inertial frame.
In this thesis, it is presented a set of results in adiabatic dynamics (closed and open system) and transitionless quantum driving that promote some advances in our understanding on quantum control and Hamiltonian inverse engineering. In the context of adiabatic dynamics in closed systems, it is introduced a validation mechanism for the adiabaticity conditions by studing the system dynamics from a non-inertial reference frame. By considering a decohering scenario, validity conditions of the adiabatic approximation are also studied. As a fresh general result with potential applications, it is shown that under decoherence the adiabaticity may still occur in the infinite time limit, as it happens for closed systems, for a class of initial quantum states. To end, the original contributions of this thesis to the theory of shortcuts to adiabaticity refers to a generalized approach of transitionless quantum driving, where one explores the gauge freedom of the quantal phase factors accompanying adiabatic trajectories. A number of theoretical applications are studied, where some theoretical prediction presented in this thesis are experimentally verified through two different experimental setups, namely a qubit encoded in the energy hyperfine structure of a Ytterbium trapped ion, and in nuclear magnetic resonance with a nuclear spin qubit.
We present a new theorem describing stable solutions for a driven quantum system. The theorem, coined `inertial theorem', is applicable for fast driving, provided the acceleration rate is small. The theorem states that in the inertial limit eigenoperators of the propagator remain invariant throughout the dynamics, accumulating dynamical and geometric phases. The proof of the theorem utilizes the structure of Liouville space and a closed Lie algebra of operators. We demonstrate the theorem by studying three explicit solutions of a harmonic oscillator, a two-level and three-level system models. These examples demonstrate that the inertial solution is superior to that obtained with the adiabatic approximation. The inertial theorem is then employed to extend the validity of the Markovian Master equation to strongly driven open quantum systems. In addition, we explore the consequence of new geometric phases associated with the driving parameters.