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arXiv:2005.03284 · Physical Review A

Bounds for nonadiabatic transitions

Takuya Hatomura, Go Kato

Published Jul 15, 2020• 6 citations
arXivHTMLPDF

Abstract

We discuss bounds for nonadiabatic transitions from the viewpoints of the adiabatic perturbation theory and the quantum speed limit. We show that the amount of nonadiabatic transitions from the $n\mathrm{th}$ level to the $m\mathrm{th}$ level is bounded by a function of the quantum geometric tensor for the $m\mathrm{th}$ level. We analyze this bound from the viewpoint of the adiabatic perturbation theory. In addition, this bound and the viewpoint of the quantum speed limit suggest a nontrivial relationship between the dynamical transformation and the adiabatic transformation. We also derive a universal bound for any nonadiabatic transition. This bound is written in terms of the counterdiabatic Hamiltonian.

DOI
10.1103/physreva.102.012216

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We discuss bounds for nonadiabatic transitions from the viewpoints of the adiabatic perturbation theory and the quantum speed limit. We show that the amount of nonadiabatic transitions from the nthn\mathrm{th}nth level to the mthm\mathrm{th}mth level is bounded by a function of the quantum geometric tensor for the mthm\mathrm{th}mth level. We analyze this bound from the viewpoint of the adiabatic perturbation theory. In addition, this bound and the viewpoint of the quantum speed limit suggest a nontrivial relationship between the dynamical transformation and the adiabatic transformation. We also derive a universal bound for any nonadiabatic transition. This bound is written in terms of the counterdiabatic Hamiltonian.
The quantum speed limit specifies a universal bound of the fidelity between the initial state and the timeevolved state.We apply this method to find a bound of the fidelity between the adiabatic state and the timeevolved state.The bound is characterized by the counterdiabatic Hamiltonian and can be used to evaluate the worst case performance of the adiabatic quantum computation.The result is improved by imposing additional conditions and we examine several models to find a tight bound.We also derive a different type of quantum speed limit that is meaningful even when we take the thermodynamic limit.By using solvable spin models, we study how the performance and the bound are affected by phase transitions.
The adiabatic control is a powerful technique for many practical applications in quantum state engineering, light-driven chemical reactions and geometrical quantum computations. This paper reveals a speed limit of nonadiabatic transition in a general time-dependent parametric quantum system that leads to an upper bound function which lays down an optimal criteria for the adiabatic controls. The upper bound function of transition rate between instantaneous eigenstates of a time-dependent system is determined by the power fluctuations of the system relative to the minimum gap between the instantaneous levels. In a parametric Hilbert space, the driving power corresponds to the quantum work done by the parametric force multiplying the parametric velocity along the parametric driving path. The general two-state time-dependent models are investigated as examples to calculate the bound functions in some general driving schemes with one and two driving parameters. The calculations show that the upper bound function provides a tighter real-time estimation of nonadiabatic transition and is closely dependent on the driving frequencies and the energy gap of the system. The deviations of the real phase from Berry phase on different closed paths are induced by the nonadiabatic transitions and can be efficiently controlled by the upper bound functions. When the upper bound is adiabatically controlled, the Berry phases of the electronic spin exhibit nonlinear step-like behaviors and it is closely related to topological structures of the complicated parametric paths on Bloch sphere.
This paper is concerned with the problem of calculating amplitudes for nonadiabatic transitions induced by a time-dependent Hamiltonian, in the semiclassical limit h/→0, with emphasis on questions relevant to semiclassical theories of electronically inelastic scattering. For this problem the semiclassical limit is mathematically equivalent to the adiabatic limit, and the adiabatic theorem says that all these transition amplitudes vanish in the limit; the question is, what is the asymptotic form of the nonadiabatic amplitudes, as they go to zero? We consider Hamiltonia that are analytic matrix functions of time. We prove a generalization of the adiabatic theorem to the complex time plane; paradoxically, the adiabatic theorem in the complex plane gives us directly the nonadiabatic amplitudes along the real time axis. We derive Dykhne’s remarkable formula for the two-state case, which says that the limiting form of the transition amplitude depends only on the energy curves of the two states, not on the nonadiabatic coupling which is responsible for transition between them. We discuss the three-state problem at length and show that the obvious generalization of the Dykhne formula is sometimes true, sometimes false. To indicate the scope of methods based on the adiabatic theorem in the complex plane, we give an elementary proof of the semiclassical formula for above-barrier reflection of a one-dimensional particle.
Applying time-dependent driving is a basic way of quantum control. Driven systems show various dynamics as its time scale is changed due to the different amount of nonadiabatic transitions. The fast-forward scaling theory enables us to observe slow (or fast) time-scale dynamics during moderate time by applying additional driving. Here we discuss its application to nonadiabatic transitions. We derive mathematical expression of additional driving and also find a formula for calculating it. Moreover, we point out relation between the fast-forward scaling theory for nonadiabatic transitions and shortcuts to adiabaticity by counterdiabatic driving.
For adiabatic controls of quantum systems, the nonadiabatic transitions are reduced by increasing the operation time of processes. Perfect quantum adiabaticity usually requires the infinitely slow variation of control parameters. In this paper, we propose the dynamical quantum geometric tensor, as a metric in the control parameter space, to speed up quantum adiabatic processes and reach quantum adiabaticity in relatively short time. The optimal protocol to reach quantum adiabaticity is to vary the control parameter with a constant velocity along the geodesic path according to the metric. For the system initiated from the nthn\mathrm{th}nth eigenstate, the transition probability in the optimal protocol is bounded by ${P}_{n}(t)\ensuremath{\le}4{\mathcal{L}}_{n}^{2}/{\ensuremath{\tau}}^{2}$ with the operation time $\ensuremath{\tau}$ and the quantum adiabatic length Ln{\mathcal{L}}_{n}Ln​ induced by the metric. Our optimization strategy is illustrated via two explicit models: the Landau-Zener model and the one-dimensional transverse Ising model.
Adiabatic gauge potential is the origin of nonadiabatic transitions. In counterdiabatic driving, which is a method of shortcuts to adiabaticity, adiabatic gauge potential can be used to realize identical dynamics to adiabatic time evolution without requiring slow change of parameters. We introduce an algebraic expression of adiabatic gauge potential. Then, we find that the explicit form of adiabatic gauge potential can be easily determined by some algebraic calculations. We demonstrate this method by using a single-spin system, a two-spin system, and the transverse Ising chain. Moreover, we derive a lower bound for fidelity to adiabatic time evolution based on the quantum speed limit. This bound enables us to know the worst case performance of approximate adiabatic gauge potential. We can also use this bound to find dominant terms in adiabatic gauge potential to suppress nonadiabatic transitions. We apply this bound to magnetization reversal of the two-spin system and to quantum annealing of the transverse Ising chain. Adiabatic gauge potential reflects structure of energy eigenstates, and thus we also discuss detection of quantum phase transitions by using adiabatic gauge potential. We find a signature of a quantum phase transition in the transverse Ising chain.