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arXiv:1706.03966 · Physical Review A

Fast forward of adiabatic control of tunneling states

Katsuhiro Nakamura, Anvar Khujakulov, S. K. Avazbaev, Shumpei Masuda

Published Jun 9, 2017• 22 citations
arXivHTMLPDF

Abstract

By developing the preceding work on the fast forward of transient phenomena of quantum tunneling by Khujakulov and Nakamura [Phys. Rev. A 93, 022101 (2016)], we propose a scheme of the exact fast forward of adiabatic control of stationary tunneling states with use of the electromagnetic field. The idea allows the acceleration of both the amplitude and phase of wave functions throughout the fast-forward time range. The scheme realizes the fast-forward observation of the transport coefficients under the adiabatically changing barrier with the fixed energy of an incoming particle. As typical examples, we choose systems with (1) Eckart's potential with tunable asymmetry and (2) double $\ensuremath{\delta}$-function barriers under tunable relative height. We elucidate the driving electric field to guarantee the stationary tunneling state during a rapid change of the barrier and evaluate both the electric-field-induced temporary deviation of transport coefficients from their stationary values and the modulation of the phase of complex scattering coefficients.

DOI
10.1103/physreva.95.062108

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By developing the preceding work on the fast forward of transient phenomena of quantum tunneling by Khujakulov and Nakamura [Phys. Rev. A 93, 022101 (2016)], we propose a scheme of the exact fast forward of adiabatic control of stationary tunneling states with use of the electromagnetic field. The idea allows the acceleration of both the amplitude and phase of wave functions throughout the fast-forward time range. The scheme realizes the fast-forward observation of the transport coefficients under the adiabatically changing barrier with the fixed energy of an incoming particle. As typical examples, we choose systems with (1) Eckart's potential with tunable asymmetry and (2) double \ensuremathδ\ensuremath{\delta}\ensuremathδ-function barriers under tunable relative height. We elucidate the driving electric field to guarantee the stationary tunneling state during a rapid change of the barrier and evaluate both the electric-field-induced temporary deviation of transport coefficients from their stationary values and the modulation of the phase of complex scattering coefficients.
We propose a scheme of the exact fast forwarding of standard quantum dynamics for a charged particle. The present idea allows the acceleration of both the amplitude and the phase of the wave function throughout the fast-forward time range and is distinct from that of Masuda and Nakamura [Proc. R. Soc. A 466, 1135 (2010)], which enabled acceleration of only the amplitude of the wave function on the way. We apply the proposed method to the quantum tunneling phenomena and obtain the electromagnetic field to ensure the rapid penetration of wave functions through a tunneling barrier. Typical examples described here are (1) an exponential wave packet passing through the $\ensuremath{\delta}$-function barrier and (2) the opened Moshinsky shutter with a $\ensuremath{\delta}$-function barrier just behind the shutter. We elucidate the tunneling current in the vicinity of the barrier and find a remarkable enhancement of the tunneling rate (tunneling power) due to the fast forwarding. In the case of a very high barrier, in particular, we present the asymptotic analysis and exhibit a suitable driving force to recover a recognizable tunneling current. The analysis is also carried out on the exact acceleration of macroscopic quantum tunneling with use of the nonlinear Schr\"odinger equation, which accommodates a tunneling barrier.
We introduce Khujakulov and Nakamura's scheme for the exact fast-forwarding of standard quantum dynamics for a charged particle.The idea allows the acceleration of both amplitude and phase of the wave function throughout the fast-forwarding time range.Firstly we shall apply the proposed method to 1-D free wave packet dynamics and obtain the electromagnetic field to ensure its rapid propagation and diffusion.Then we proceed to study 1-D quantum tunneling phenomenon, namely a rapid penetration of wave function through a delta-function type barrier.We elucidate the distribution of the tunneling current density to show the remarkable enhancement of the tunneling rate (tunneling power) due the fast-forwarding.We introduce two types of time-magnification factors and confirm the stability of fast-forward against the variation of such factors.
We propose a method to accelerate adiabatic dynamics of wave functions (WFs) in quantum mechanics to obtain a final adiabatic state except for the spatially uniform phase in any desired short time. In our previous work, acceleration of the dynamics of WFs was shown to obtain the final state in any short time by applying driving potential. We develop the previous theory of fast-forward to derive a driving potential for the fast-forward of adiabatic dynamics. A typical example is the fast-forward of adiabatic transport of a WF, which is the ideal transport in the sense that a stationary WF is transported to an aimed position in any desired short time without leaving any disturbance at the final time of the fast-forward. As other important examples, we show accelerated manipulations of WFs, such as their splitting and squeezing. The theory is also applicable to macroscopic quantum mechanics described by the nonlinear Schrödinger equation.
We study a scheme of fast-forward adiabatic quantum dynamics of a (2 + 1) Dirac particle. This scheme was originally proposed by Masuda and Nakamura. In this scheme, we include the adiabatic parameter that maintains the adiabatic motion of the particle and fast forward its motion by introducing a time scaling parameter. The fast forward adiabatic state is obtained by determining the regularization term and driving potential. We introduce the proposed method to the system with the Dirac particle using a (2 + 1) dimensiontime- dependent Dirac equation and obtain the regularization term, the driving scalar potential VFF and the driving vector potential AFF . By tuning the driving electric field, this method can accelerate the adiabatic dynamics of an electron as a Dirac particle trapped in the ground state in the plane xy and an electric field in the x direction and a constant magnetic field in the y direction. This acceleration will preserve the ground state of the wave function from the initial time to the final time.
We show a method to accelerate quantum adiabatic dynamics of wave functions under electromagnetic field (EMF) by developing the preceding theory [Masuda and Nakamura, Proc. R. Soc. London Ser. A 466, 1135 (2010)]. Treating the orbital dynamics of a charged particle in EMF, we derive the driving field which accelerates quantum adiabatic dynamics in order to obtain the final adiabatic states in any desired short time. The scheme is consolidated by describing a way to overcome possible singularities in both the additional phase and driving potential due to nodes proper to wave functions under EMF. As explicit examples, we exhibit the fast forward of adiabatic squeezing and transport of excited Landau states with nonzero angular momentum, obtaining the result consistent with the transitionless quantum driving applied to the orbital dynamics in EMF.
We show a method to accelerate quantum adiabatic dynamics of wavefunctions under electro-magnetic field by developing the previous theory (Masuda & Nakamura 2008 and 2010). Firstly we investigate the orbital dynamics of a charged particle. We derive the driving field which accelerates quantum adiabatic dynamics in order to obtain the final adiabatic states except for the spatially uniform phase such as the adiabatic phase in any desired short time. Fast-forward of adiabatic squeezing and transport in the electro-magnetic field is exhibited. Secondly we investigate spin dynamics under the magnetic field, showing the fast-forward of adiabatic spin inversion and of adiabatic dynamics in Landau-Zener model. The connection of the present framework with Kato-Berry's transitionless quantum driving is elucidated in Appendix.