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arXiv:hep-th/9704062 · Physical Review A

Duality in potential curve crossing: Application to quantum coherence

Kazuo Fujikawa, Hiroshi Suzuki

Published Nov 1, 1997• 3 citations
arXivHTMLPDF

Abstract

A field-dependent su(2) gauge transformation connects between the adiabatic and diabatic pictures in the (Landau-Zener-Stueckelberg) potential curve crossing. It is pointed out that weak and strong potential curve crossing interactions are interchanged under this transformation, thus realizing a naive strong and weak duality. A reliable perturbation theory should thus be formulated in limits of both weak and strong interactions. In fact, the main characteristics of the potential crossing phenomena such as the Landau-Zener formula including its numerical coefficient are well described by simple (time-independent) perturbation theory without referring to Stokes phenomena. We also show that quantum coherence in a double-well potential is generally suppressed by the effect of a potential curve crossing, which is analogous to the effect of Ohmic dissipation on quantum coherence.

DOI
10.1103/physreva.56.3436

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Abstract We study the dynamics of a quantum two-state system driven through an avoided crossing under the influence of a super-Ohmic environment. We determine the Landau–Zener probability employing the numerical exact quasi-adiabatic path integral and a Markovian weak coupling approach. Increasing the driving time in the numerical protocol, we find converged results which shows that super-Ohmic environments only influence the Landau Zener probability within a finite crossing time window. This crossing time is qualitatively determined by the environmental cut-off energy. At weak coupling, we show that the Markovian weak coupling approach provides an accurate description. Since pure dephasing of a super-Ohmic bath is non-Markovian, this highlights that pure dephasing hardly influences the Landau–Zener probability. The finite crossing time window, thus, results from the suppression of relaxation once the energy splitting exceeds the environmental cut-off energy. Graphical abstract

The European Physical Journal B•Mar 1, 2022•1 citation
A field-dependent su(2) gauge transformation connects between the adiabatic and diabatic pictures in the (Landau-Zener-Stueckelberg) potential curve crossing. It is pointed out that weak and strong potential curve crossing interactions are interchanged under this transformation, thus realizing a naive strong and weak duality. A reliable perturbation theory should thus be formulated in limits of both weak and strong interactions. In fact, the main characteristics of the potential crossing phenomena such as the Landau-Zener formula including its numerical coefficient are well described by simple (time-independent) perturbation theory without referring to Stokes phenomena. We also show that quantum coherence in a double-well potential is generally suppressed by the effect of a potential curve crossing, which is analogous to the effect of Ohmic dissipation on quantum coherence.
It is pointed out that there exists an interesting strong and weak duality in the Landau-Zener-Stueckelberg potential curve crossing. A reliable perturbation theory can thus be formulated in the both limits of weak and strong interactions. It is shown that main characteristics of the potential crossing phenomena such as the Landau-Zener formula including its numerical coefficient are well-described by simple (time-independent) perturbation theory without referring to Stokes phenomena. A kink-like topological object appears in the ``magnetic'' picture, which is responsible for the absence of the coupling constant in the prefactor of the Landau-Zener formula. It is also shown that quantum coherence in a double well potential is generally suppressed by the effect of potential curve crossing, which is analogous to the effect of Ohmic dissipation on quantum coherence.
The compact analytical complete solutions recently obtained for the two-state Landau-Zener-Stueckelberg problems are reviewed and explained. The theory covers both Landau-Zener (LZ) type in which the two diabatic potential curves cross with the same sign of slopes and the nonadiabatic tunneling (NT) type in which the potentials cross with different signs of slopes. The theory is applicable virtually in the whole range of energy and coupling strength and is convenient for practical use. The new theory for time-dependent nonadiabatic transition can be formulated from the time-independent theory of the LZ-type. The utilizability of the theory to various multi-channel problems and also to multi-dimensional problems is demonstrated and explained. The intriguing phenomenon of complete reflection which appears in the NT-case is explicitly utilized to propose a new type of molecular switching and to control molecular processes such as molecular photodissociation. Also proposed is a new way of controlling molecular processes by using time-dependent external fields. Finally, a trial to formulate a unified analytical theory to cover both Landau-Zener-Stueckelberg and Rosen-Zener-Demkov types of nonadiabatic transitions is briefly touched upon.
Landau-Zener tunneling, which describes the transition in a two-level system during a sweep through an anti-crossing, is a model applicable to a wide range of physical phenomena. Realistic quantum systems are affected by dissipation due to coupling to their environments. An important aspect of understanding such open quantum systems is the relative energy scales of the system itself and the system-environment coupling, which distinguishes the weak- and strong-coupling regimes. Using a tunable superconducting flux qubit, we observe the crossover from weak to strong coupling to the environment in Landau-Zener tunneling. Our results confirm previous theoretical studies of dissipative Landau-Zener tunneling in the weak and strong coupling limits. We devise a spin bath model that effectively captures the crossover regime. This work is relevant for understanding the role of dissipation in quantum annealing, where the system is expected to go through a cascade of Landau-Zener transitions before reaching the target state.
Based on the achievements of the previous three papers of this series, the best working formulas for scattering matrix are obtained for both the Landau–Zener (LZ) and the nonadiabatic tunneling (NT) case: two formulas valid at b2≥0 and b2≤0 in the LZ case, and three formulas valid atb2≤−1, −1≤b2≤1 and b2≥1 in the NT case, where b2 represents the effective energy. Simple and compact formulas which work far better than the LZ formula are proposed for nonadiabatic transition probability by one passage of crossing point for both the LZ and NT cases. Furthermore, compact expressions are derived, for the first time, for the nonadiabatic tunneling probability at b2≤1, i.e., at energies lower than the bottom of the upper adiabatic potential. All the formulas proposed here can be usefully utilized at any coupling strength, namely the validity range has been very much expanded compared to the previous formulas by employing certain empirical corrections. Besides, these formulas have convenience to enable an extension to general curved potentials.
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.
Abstract We study the dynamics of a quantum two-state system driven through an avoided crossing under the influence of a super-Ohmic environment. We determine the Landau–Zener probability employing the numerical exact quasi-adiabatic path integral and a Markovian weak coupling approach. Increasing the driving time in the numerical protocol, we find converged results which shows that super-Ohmic environments only influence the Landau Zener probability within a finite crossing time window. This crossing time is qualitatively determined by the environmental cut-off energy. At weak coupling, we show that the Markovian weak coupling approach provides an accurate description. Since pure dephasing of a super-Ohmic bath is non-Markovian, this highlights that pure dephasing hardly influences the Landau–Zener probability. The finite crossing time window, thus, results from the suppression of relaxation once the energy splitting exceeds the environmental cut-off energy. Graphical abstract