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Science Technology and Engineering

The Study of Single-phonon Landau-Zener Tunneling Pobability

Cheng Shao-rong

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

A new complete basis set was adopted to solve single-phonon Landau-Zener tunneling probability by solving the Schrodinger differential equation,then the result was compared with the result got by solving Landau-Zener formula.The result show that:In the case of weak coupling,the tunneling probability by solving the Schrodinger differential equation is consistent with the result got by solving Landau-Zener formula.

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A new complete basis set was adopted to solve single-phonon Landau-Zener tunneling probability by solving the Schrodinger differential equation,then the result was compared with the result got by solving Landau-Zener formula.The result show that:In the case of weak coupling,the tunneling probability by solving the Schrodinger differential equation is consistent with the result got by solving Landau-Zener formula.
We obtain probabilities at a crossing of two linearly time-dependent potentials that are constantly coupled to the other by solving a time-dependent Schrödinger equation. We find that the system which was initially localized at one state evolves to split into both states at the crossing. The probability splitting depends on the coupling strength V0V_0V0​ such that the system stays at the initial state in its entirety when V0V_0V0​ = 0 while it is divided equally in both states when V0→∞V_0 \rightarrow {\infty}V0​→∞ . For a finite coupling the probability branching at the crossing is not even and thus a complete probability transfer at t→∞t \rightarrow {\infty}t→∞ is not achieved in the linear potential crossing problem. The Landau-Zener formula for transition probability at t→∞t \rightarrow {\infty}t→∞ is expressed in terms of the probabilities at the crossing.
To determine the possibility of single-electron operation in electronic devices, we investigated the Landau-Zener (LZ) tunneling process with dissipation. This work presents an analytical formula and detailed numerical results for dynamical tunneling phenomena with an Ohmic reservoir and a noise source. First, we calculated the transient and final density matrices of a single-electron system coupled to a reservoir at zero temperature, and then noted the rigid tunneling process and a noticeable decoherence of the tunneling. The rigidity of the LZ tunneling suggests that we can suppress errors in single-electron operations. However, our analytical and numerical results clarified the fact that noise strongly affects the tunneling process. When half of the noise spectrum bandwidth ${\ensuremath{\omega}}_{\mathrm{max}}$ is larger than the gap of the adiabatic potential, thermal excitation by the noise source considerably increases the tunneling probability. On the other hand, our numerical results found that the noise effect is proportional to ${\ensuremath{\omega}}_{\mathrm{max}}^{3}$ when ${\ensuremath{\omega}}_{\mathrm{max}}$ is smaller than the gap. Then we can accurately operate single electrons if the noise source has a sufficiently narrow band.
We present a rigorous analysis of the generalized Landau-Zener problem for the two-level interacting Bose-Einstein condensates. We show that the dynamics of the system is accurately, in detail, described by a two-term variational ansatz that is valid for the whole time domain and is applicable for any set of involved parameters. Applying an exact third-order nonlinear differential equation we construct an advanced fifth-order polynomial equation for the final transition probability serving as a highly accurate generalized Landau-Zener formula.
We study Landau-Zener transitions in a qubit coupled to a bath at zero temperature. A general formula that is applicable to models with a nondegenerate ground state is derived. We calculate exact transition probabilities for a qubit coupled to either a bosonic or a spin bath. The nature of the baths and the qubit-bath coupling is reflected in the transition probabilities. For diagonal coupling, when the bath causes energy fluctuations of the diabatic qubit states but no transitions between them, the transition probability coincides with the standard Landau-Zener probability of an isolated qubit. This result is universal as it does not depend on the specific type of bath. For pure off-diagonal coupling, by contrast, the tunneling probability is sensitive to the coupling strength. We discuss the relevance of our results for experiments on molecular nanomagnets, in circuit QED, and for the fast-pulse readout of superconducting phase qubits.
In the framework of a basic semiclassical time-dependent nonlinear two-state problem, we study the weak coupling limit of the nonlinear Landau-Zener transition at coherent photo- and magneto-association of an atomic Bose-Einstein condensate. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct an accurate analytic approximation describing time dynamics of the coupled atom-molecular system for the case of weak coupling. The approximation is written in terms of the solution to an auxiliary linear Landau-Zener problem with some effective Landau-Zener parameter. The dependence of this effective parameter on the input Landau-Zener parameter is found to be unexpected: as the generic Landau-Zener parameter increases, the effective Landau-Zener parameter first monotonically increases (starting from zero), reaches its maximal value and then monotonically decreases again reaching zero at some point. The constructed approximation quantitatively well describes many characteristics of the time dynamics of the system, in particular, it provides a highly accurate formula for the final transition probability to the molecular state. The present result for the final transition probability improves the accuracy of the previous approximation by Ishkhanyan et al. [Phys. Rev. A 69 , 043612 (2004); J. Phys. A 38 , 3505 (2005)] by order of magnitude.
The basic properties of the nonlinear Landau-Zener tunnelling,a new quantum tunnelling phenomenon is reviewed,which is discovered in interacting two-level systems,such as Bose-Einstein condensates in optical lattices and molecule magnets.It is found that the nonlinearity induced by the repulsive interaction can enhance the tunnelling probability compared to the usual linear Landdau-Zener tunnelling.When the nonlinearity is strong enough,the adiabaticity can break down:an infinitely small external linear force can cause finite tunnelling probability.