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arXiv:1807.10227 · Physical Review A

Fast adiabatic evolution by oscillating initial Hamiltonians

Francesco Petiziol, Benjamin Dive, Florian Mintert, Sandro Wimberger

Published Oct 31, 2018• 58 citations
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Abstract

We propose a method to produce fast transitionless dynamics for finite-dimensional quantum systems without requiring additional Hamiltonian components not included in the initial control setup, remaining close to the true adiabatic path at all times. The strategy is based on the introduction of an effective counterdiabatic scheme: a correcting Hamiltonian is constructed which approximatively cancels nonadiabatic effects, inducing an evolution tracking the adiabatic states closely. This can be absorbed into the initial Hamiltonian by adding a fast oscillation in the control parameters. We show that a consistent speedup can be achieved without requiring strong control Hamiltonians, using it both as a stand-alone shortcut to adiabaticity and as a weak correcting field. A number of examples are treated, dealing with quantum state transfer in avoided-crossing problems and entanglement creation.

DOI
10.1103/physreva.98.043436

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We propose a method to produce fast transitionless dynamics for finite-dimensional quantum systems without requiring additional Hamiltonian components not included in the initial control setup, remaining close to the true adiabatic path at all times. The strategy is based on the introduction of an effective counterdiabatic scheme: a correcting Hamiltonian is constructed which approximatively cancels nonadiabatic effects, inducing an evolution tracking the adiabatic states closely. This can be absorbed into the initial Hamiltonian by adding a fast oscillation in the control parameters. We show that a consistent speedup can be achieved without requiring strong control Hamiltonians, using it both as a stand-alone shortcut to adiabaticity and as a weak correcting field. A number of examples are treated, dealing with quantum state transfer in avoided-crossing problems and entanglement creation.
Transitionless quantum driving achieves adiabatic evolution in a hurry, using a counterdiabatic Hamiltonian to stifle nonadiabatic transitions. Here this shortcut to adiabaticity is cast in terms of a generator of adiabatic transport. This yields a classical analog of transitionless driving, and provides a strategy for constructing quantal counterdiabatic Hamiltonians. As an application of this framework, exact classical and quantal counterdiabatic terms are obtained for a particle in a box and for even-power-law potentials in one degree of freedom.
We propose a scheme to speed up adiabatic passage by using Lyapunov control theory. This is a good choice to solve the problem that may emerge in Berry's transitionless quantum driving [M. V. Berry, J. Phys. A 42, 365303 (2009)]. That is, the extra couplings in the counterdiabatic driving Hamiltonian can be avoided by choosing the available control Hamiltonian in an actual physical system. As examples, we shorten the evolution time of adiabatic population transfer in a three-level system and the entanglement generation in a cavity quantum electrodynamics system. Moreover, the occupation of an intermediate state can be sharply suppressed by properly choosing the control Hamiltonian in the three-level system. The scheme can also be generalized to a complex system where the exact expressions of adiabatic eigenstates are difficult to obtain.
The challenge to fully grasp and to harness the bizarre phenomena which take place in the quantum realm is crucially linked to the ability to engineer and manipulate quantum systems with high precision. The scientific progress in this direction is opening new routes for the investigation of fundamental quantum physics, and it is at the same time paving the way to the development of the first quantum technologies. \n\nThis thesis is dedicated to the search of strategies for high-fidelity and noise-tolerant quantum control.\nA versatile theoretical framework is set up for the design of optimized control protocols based on accelerated adiabatic processes. The benefits of the adiabatic dynamics, rooted in its intrinsic stability versus control imperfections, are released from the requirement of slow evolution: this is achieved thanks to the theoretical construction of accelerating control Hamiltonians which dynamically counteract deviations from the target dynamics. Our framework provides a methodical way to engineer these Hamiltonians by introducing fast modulations of the initially-available control parameters, and it results from the interplay between the theory of counterdiabatic fields and Floquet-Magnus average Hamiltonian theory. As compared to other shortcut-to-adiabaticity strategies, our control method keeps the system always close to the exact adiabatic path while not requiring the implementation of new Hamiltonian couplings. Moreover, the flexibility in the construction of the accelerating Hamiltonians offers a fertile ground for the hybridization with optimal control techniques.\n\nThe general setup is applied to exemplary control problems of relevance in quantum optics and quantum information processing. First, a protocol is designed for the generation of entanglement between effective two-level systems coupled through a transmission line, in the experimental context of circuit quantum electrodynamics. Second, an accelerated version of the widespread technique of stimulated Raman adiabatic passage is presented, which dramatically enlarges its range of applicability for implementations both in the optical and in the microwave regime.\n\nThe results strongly support the potential of the quantum control framework developed in this thesis. In particular, our methodology offers the opportunity to flank high fidelities and small error sensitivity with on-demand adaptability to experimental constraints and the flexibility for integrations with other control techniques. This makes it a valuable resource for applications in many branches of quantum science, with a natural outlet in quantum computation and simulation.
Abstract We review a scheme for the systematic design of quantum control protocols based on shortcuts to adiabaticity in few-level quantum systems. The adiabatic dynamics is accelerated by introducing high-frequency modulations in the control Hamiltonian, which mimic a time-dependent counterdiabatic correction. We present a number of applications for the high-fidelity realization of quantum state transfers and quantum gates based on effective counterdiabatic driving, in platforms ranging from superconducting circuits to Rydberg atoms.
The adiabatic theorem and shortcuts to adiabaticity for time-dependent open quantum systems are explored in this paper. Starting from the definition of dynamical stable decoherence-free subspace, we show that, under a compact adiabatic condition, the quantum state remains in the time-dependent decoherence-free subspace with an extremely high purity, even though the dynamics of the open quantum system may not be adiabatic. The adiabatic condition mentioned here in the adiabatic theorem for open systems is very similar to that for closed quantum systems, except that the operators required to change slowly are the Lindblad operators. We also show that the adiabatic evolution of decoherence-free subspaces depends on the existence of instantaneous decoherence-free subspaces, which requires that the Hamiltonian of open quantum systems be engineered according to the incoherent control protocol. In addition, shortcuts to adiabaticity for adiabatic decoherence-free subspaces are also presented based on the transitionless quantum driving method. Finally, we provide an example that consists of a two-level system coupled to a broadband squeezed vacuum field to show our theory. Our approach employs Markovian master equations and the theory can apply to finite-dimensional quantum open systems.
A shortcut to adiabaticity is a recipe for generating adiabatic evolution at an arbitrary pace. Shortcuts have been developed for quantum, classical and (most recently) stochastic dynamics. A shortcut might involve a counterdiabatic (CD) Hamiltonian that causes a system to follow the adiabatic evolution at all times, or it might utilize a fast-forward (FF) potential, which returns the system to the adiabatic path at the end of the process. We develop a general framework for constructing shortcuts to adiabaticity from flow fields that describe the desired adiabatic evolution. Our approach encompasses quantum, classical and stochastic dynamics, and provides surprisingly compact expressions for both CD Hamiltonians and FF potentials. We illustrate our method with numerical simulations of a model system, and we compare our shortcuts with previously obtained results. We also consider the semiclassical connections between our quantum and classical shortcuts. Our method, like the FF approach developed by previous authors, is susceptible to singularities when applied to excited states of quantum systems; we propose a simple, intuitive criterion for determining whether these singularities will arise, for a given excited state.