Adiabatic processes driven by non-Hermitian, time-dependent Hamiltonians may be sped up by generalizing inverse engineering techniques based on counter-diabatic (transitionless driving) algorithms or on dynamical invariants. We work out the basic theory and examples described by two-level Hamiltonians: the acceleration of rapid adiabatic passage with a decaying excited level and of the dynamics of a classical particle on an expanding harmonic oscillator.
Adiabatic processes driven by non-Hermitian, time-dependent Hamiltonians may be sped up by generalizing inverse engineering techniques based on counter-diabatic (transitionless driving) algorithms or on dynamical invariants. We work out the basic theory and examples described by two-level Hamiltonians: the acceleration of rapid adiabatic passage with a decaying excited level and of the dynamics of a classical particle on an expanding harmonic oscillator.
By definition, quantum adiabatic process means a slow process which allows the state to follow the instantaneous eigenstates of time-dependent Hamiltonian. For some reasons, for example, to avoid the decoherence effects, or to implement large-scale quantum information processing, we may want to speed up the adiabatic process, that is, to achieve the adiabatic-like control but in very short time. In this talk, we present new methods to speeded-up expansion and transport of cold atoms in a harmonic trap. Such shortcuts to adiabaticity provide the fast expansion or transport, in which the final atomic state is the same as in the adiabatic process, but in arbitrarily short time, keeping the same populations of vibrational levels in the initial and final trap. These methods can also be generalized to Bose-Einstein Condensates. Moreover, we present shortcuts to adiabatic passage from one internal atomic state to another in twoor three-level atoms. In detail, we can apply Berry’s transitionless quantum driving method and invariant-based inverse engineering method to accelerate the Rapid Adiabatic Passage (RAP) and Stimulated Raman Adiabatic Passage (STIRAP) in twoand three-level atomic systems, respectively. We also examine and compare the stability of different schemes concerning different types of noise and system errors. Finally, we show that the technique of shortcuts to adiabaticity in two-level system can be applied in fast and robust spin flip in a quantum dot, to avoid the decoherent effects.
Different methods have been recently put forward and implemented experimentally to inverse engineer the time-dependent Hamiltonian of a quantum system and accelerate slow adiabatic processes via nonadiabatic shortcuts. In the ``transitionless quantum driving'' proposed by Berry, shortcut Hamiltonians are designed so that the system follows exactly, in an arbitrarily short time, the approximate adiabatic path defined by a reference Hamiltonian. A different approach is based on first designing a Lewis-Riesenfeld invariant to carry the eigenstates of a Hamiltonian from specified initial to final configurations, again in an arbitrary time, and then constructing from the invariant the transient Hamiltonian that connects these boundary configurations. We show that the two approaches, apparently quite different in form and so far in results, are, in fact, strongly related and potentially equivalent, so that the inverse-engineering operations in one of them can be reinterpreted and understood in terms of the concepts and operations of the other one. We study, as explicit examples, expansions of time-dependent harmonic traps and the state preparation of two-level systems.
We use the dynamical algebra of a quantum system and its dynamical invariants to inverse engineer feasible Hamiltonians for implementing shortcuts to adiabaticity. These are speeded up processes that end up with the same populations as slow, adiabatic ones. As application examples, we design families of shortcut Hamiltonians that drive two- and three-level systems between initial and final configurations, imposing physically motivated constraints on the terms (generators) allowed in the Hamiltonian.
Shortcuts to adiabaticity are alternative fast processes which reproduce the same final state as the adiabatic process in a finite or even shorter time, which have been extended from Hermitian systems to non-Hermitian systems in recent years, but they are barely explored for general non-Hermitian systems where off-diagonal elements of the Hamiltonian are not Hermitian. In this paper, we propose a shortcuts to adiabaticity technique which is based on a transitionless quantum driving algorithm to realize population transfer for general two-level non-Hermitian systems and give both exact and approximate analytical solutions of the corresponding counteradiabatic driving Hamiltonian, where the latter can be extended to the zeroth-order and first-order terms by applying perturbative theory. We find that the first-order correction term is different from the previous results, which is caused by the non-Hermiticity of the off-diagonal elements. We work out an exact expression for the control function and present examples consisting of a general two-level system with gain and loss to show the theory. The results suggest that the high-fidelity population transfer can be implemented in general non-Hermitian systems by our method, which works even with strong non-Hermiticity and without rotating wave approximation. Furthermore, we show that the general Hamiltonian the off-diagonal elements of which are not conjugate to each other can be implemented in many physical systems with the present experimental technology, such as an atom-light interaction system and whispering-gallery microcavity, which might have potential applications in quantum information processing.
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 propose an efficient method to construct shortcuts to adiabaticity through designing a substitute Hamiltonian to try to avoid the defect in which the speed-up protocols' Hamiltonian may involve terms which are difficult to realize in practice. We show that as long as the counterdiabatic coupling terms---even only some of them---have been nullified by the additional Hamiltonian, the corresponding shortcuts to the adiabatic process could be constructed and the adiabatic process would be sped up. As an application example, we apply this method to the popular Landau-Zener model for the realization of fast population inversion. The results show that in both Hermitian and non-Hermitian systems, we can design different additional Hamiltonians to replace the traditional counterdiabatic driving Hamiltonian to speed up the process. This method provides many choices for designing additional terms of the Hamiltonian such that one can choose a realizable model in practice.