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Physical Review A

Landau-Zener formula for the adiabatic gauge potential

Gabriel Cardoso

Published Feb 1, 2024• 4 citations
Publisher

Abstract

We study the stability of counterdiabatic drive by computing the joint dependence of the probability of nonadiabatic transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ on the adiabatic parameter $\ensuremath{\delta}$ and on the normalized amplitude of the adiabatic gauge potential (AGP) $\ensuremath{\eta}$ in the Landau-Zener-St\"uckelberg-Majorana model. We show that the Dykhne-Davis-Pechukas formula cannot be readily applied since the AGP introduces a singularity in the Hamiltonian which makes the wave function multivalued in the complex-time plane. This can be understood as the non-Abelian Aharonov-Bohm phase introduced by the AGP and leads to the counterdiabatic correction of the Landau-Zener formula. In particular, it shows that, unlike the nonperturbative suppression of transitions in the adiabatic limit $\ensuremath{\delta}\ensuremath{\rightarrow}0$, the probability is only perturbatively suppressed in the counterdiabatic limit $\ensuremath{\eta}\ensuremath{\rightarrow}1$. We then consider the extension of our results to integrable time-dependent quantum Hamiltonians. We prove that the AGP satisfies the flatness constraint which characterizes integrability in these models, which allows us to derive simple expressions for the AGP and the probability of transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ near adiabatic or counterdiabatic evolution in three- and four-state integrable examples.

DOI
10.1103/physreva.109.022201

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We study the stability of counterdiabatic drive by computing the joint dependence of the probability of nonadiabatic transitions P\ensuremathη(\ensuremathδ){P}_{\ensuremath{\eta}}(\ensuremath{\delta})P\ensuremathη​(\ensuremathδ) on the adiabatic parameter \ensuremathδ\ensuremath{\delta}\ensuremathδ and on the normalized amplitude of the adiabatic gauge potential (AGP) \ensuremathη\ensuremath{\eta}\ensuremathη in the Landau-Zener-St"uckelberg-Majorana model. We show that the Dykhne-Davis-Pechukas formula cannot be readily applied since the AGP introduces a singularity in the Hamiltonian which makes the wave function multivalued in the complex-time plane. This can be understood as the non-Abelian Aharonov-Bohm phase introduced by the AGP and leads to the counterdiabatic correction of the Landau-Zener formula. In particular, it shows that, unlike the nonperturbative suppression of transitions in the adiabatic limit \ensuremathδ\ensuremath→0\ensuremath{\delta}\ensuremath{\rightarrow}0\ensuremathδ\ensuremath→0, the probability is only perturbatively suppressed in the counterdiabatic limit \ensuremathη\ensuremath→1\ensuremath{\eta}\ensuremath{\rightarrow}1\ensuremathη\ensuremath→1. We then consider the extension of our results to integrable time-dependent quantum Hamiltonians. We prove that the AGP satisfies the flatness constraint which characterizes integrability in these models, which allows us to derive simple expressions for the AGP and the probability of transitions P\ensuremathη(\ensuremathδ){P}_{\ensuremath{\eta}}(\ensuremath{\delta})P\ensuremathη​(\ensuremathδ) near adiabatic or counterdiabatic evolution in three- and four-state integrable examples.
By the adiabatic theorem, the probability of non-adiabatic transitions in a time-dependent quantum system vanishes in the adiabatic limit. The Landau-Zener (LZ) formula gives the leading functional behavior of the probability close to this limit. On the other hand, in counterdiabatic dynamics, one achieves effectively adiabatic evolution at finite driving speed by adding an extra field which suppresses non-adiabatic transitions: the adiabatic gauge potential (AGP). We investigate the mechanism by which the AGP suppresses the transition probability, changing it from the LZ formula to exactly zero. Quantitatively, we find that adding the AGP to the Hamiltonian modifies the LZ formula by a universal prefactor, independent of the adiabatic parameter, which vanishes in the counterdiabatic regime. Qualitatively, this prefactor can be understood as arising from the Aharonov-Bohm phases generated by the AGP between different paths in the complex time plane. Finally, we show that these results extend to a class of integrable time-dependent quantum Hamiltonians by proving that the AGP preserves their integrability condition.
The approach by Ettore Majorana for non-adiabatic transitions between two quasi-crossing levels is revisited and significantly extended. We rederive the transition probability, known as the Landau-Zener-Stückelberg-Majorana formula, and introduce Majorana's approach to modern readers. This result, typically referred as the Landau-Zener formula, was published by Majorana before Landau, Zener and Stückelberg. Moreover, we go well beyond previous results and we now obtain the full wave function, including its phase, which is important nowadays for quantum control and quantum information. The asymptotic wave function correctly describes the dynamics away from the avoided-level crossing, while it has limited accuracy in that region.
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.
The control of nonadiabatic quantum transitions is a requirement for the development of modern quantum technologies. The paradigm to understand these dynamics is the Landau-Zener model. However, for nearly a century, the exact transition probability of this model has been derived using special functions or semiclassical approximations. These conventional methods obscure the underlying physical mechanisms and transition dynamics near the avoided crossing. This cumulative thesis, resulting from the author's research activities at the Intitute of Quantum Physics at Ulm University under the supervision of Prof. Dr. Wolfgang P. Schleich, establishes an analytical framework to study the underlying structure of two-level crossing dynamics. In contrast to the standard techniques, the framework developed in this thesis is based on elementary concepts combined with complex analysis. First, we map the Landau-Zener equations onto an integro-differential equation to decouple them and formulate the Markov approximation which reduces the complicated integrals to the evaluation of Fresnel integrals. This technique reproduces in the asymptotic limit the exact probability amplitude of the Landau-Zener model. Next, to uncover the origin behind the success of the Markov approximation, we analyze the Landau-Zener equations in terms of their underlying nonlinear dynamics. First, we introduce an amplitude-and-phase approach, which explains in a clear way the emergence of the Stueckelberg oscillations through the dynamics of the phase velocity. Second, we map the two-level dynamics onto a nonlinear Riccati differential equation and show that the Markov approximation is equivalent to a linearization of the derived Riccati equation. Moreover, our approach helps to identify the nonlinear term as the reason for the breakdown of the Markov approximation close to the avoided crossing. Finally, we derive exact asymptotic expressions for the transition amplitudes using elementary functions for both negative and positive times. Connecting these two regimes via the analytical continuation of the logarithmic phase reveals that the Landau-Zener transition is mainly governed by a logarithmic phase singularity. Based on this logarithmic phase singularity, we introduce a new time scale to map the system dynamics as a path via a logarithmic phase ansatz in the complex plane. Evaluating this path via the Cauchy’s integral formula yields the exact transition probability which demonstrates that the exact solution emerges from the underlying phase singularity rather than properties of special functions.
We examine the effect of nonlinearity at a level crossing on the probability for nonadiabatic transitions P. By using the Dykhne-Davis-Pechukas formula, we derive simple analytic estimates for P for two types of nonlinear crossings. In the first type, the nonlinearity in the detuning appears as a perturbative correction to the dominant linear time dependence. Then appreciable deviations from the Landau-Zener probability PLZ{P}_{\mathrm{LZ}}PLZ​ are found to appear for large couplings only, when P is small; this explains why the Landau-Zener model is often seen to provide more accurate results than expected. In the second type of nonlinearity, called essential nonlinearity, the detuning is proportional to an odd power of time. Then the nonadiabatic probability P is qualitatively and quantitatively different from PLZ{P}_{\mathrm{LZ}}PLZ​ because, on the one hand, it vanishes in an oscillatory manner as the coupling increases, and, on the other, it is much larger than PLZ.{P}_{\mathrm{LZ}}.PLZ​. We suggest experimental situations where the predicted deviations can be observed.
The Landau--Zener (LZ) model describes a two-level quantum system that undergoes an avoided crossing. In the adiabatic limit, the transition probability vanishes. An auxiliary control field HCDH_\text{CD}HCD​ can be reverse-engineered so that the full Hamiltonian H0+HCDH_0 + H_\text{CD}H0​+HCD​ reproduces adiabaticity for all parameter values. Our aim is to construct a single control field H1H_1H1​ that drives an ensemble of LZ-type Hamiltonians with a distribution of energy gaps. H1H_1H1​ works best statistically, minimizing the average transition probability. We restrict our attention to a special class of H1H_1H1​ controls, motivated by HCDH_\text{CD}HCD​. We found a systematic trade-off between instantaneous adiabaticity and the final transition probability. Certain limiting cases with a linear sweep can be treated analytically; one of them being the LZ system with Dirac δ(t)δ(t)δ(t) function. Comprehensive and systematic numerical simulations support and extend the analytic results.