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arXiv:2003.06940 · Physical Review A

Transient quantum beats, Rabi oscillations, and delay time of modulated matter waves

Jorge Villavicencio, Alberto Hernández-Maldonado

Published Apr 14, 2020• 8 citations
arXivHTMLPDF

Abstract

Transient phenomena of phase-modulated cutoff wave packets are explored by deriving an exact general solution to Schr\"odinger's equation for finite-range potentials involving arbitrary initial quantum states. We show that the dynamical features of the probability density are governed by a virtual self-induced two-level system with energies ${E}_{+}$ and ${E}_{\ensuremath{-}}$ due to the phase modulation of the initial state. The asymptotic probability density exhibits Rabi oscillations characterized by the frequency $\mathrm{\ensuremath{\Omega}}=({E}_{+}\ensuremath{-}{E}_{\ensuremath{-}})/\ensuremath{\hbar}$, which are independent of the potential profile. It is also found that for a system with a bound state, the interplay between the virtual levels with the latter causes a quantum beat effect with a beating frequency, $\mathrm{\ensuremath{\Omega}}$. We also find a regime characterized by a time-diffraction phenomenon that allows us to measure unambiguously the delay time, which can be described by an exact analytical formula. It is found that the delay time agrees with the phase time only for the case of strictly monochromatic waves.

DOI
10.1103/physreva.101.042109

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Transient phenomena of phase-modulated cutoff wave packets are explored by deriving an exact general solution to Schr"odinger's equation for finite-range potentials involving arbitrary initial quantum states. We show that the dynamical features of the probability density are governed by a virtual self-induced two-level system with energies E+{E}_{+}E+​ and E\ensuremath−{E}_{\ensuremath{-}}E\ensuremath−​ due to the phase modulation of the initial state. The asymptotic probability density exhibits Rabi oscillations characterized by the frequency \ensuremathΩ=(E+\ensuremath−E\ensuremath−)/\ensuremathℏ\mathrm{\ensuremath{\Omega}}=({E}_{+}\ensuremath{-}{E}_{\ensuremath{-}})/\ensuremath{\hbar}\ensuremathΩ=(E+​\ensuremath−E\ensuremath−​)/\ensuremathℏ, which are independent of the potential profile. It is also found that for a system with a bound state, the interplay between the virtual levels with the latter causes a quantum beat effect with a beating frequency, \ensuremathΩ\mathrm{\ensuremath{\Omega}}\ensuremathΩ. We also find a regime characterized by a time-diffraction phenomenon that allows us to measure unambiguously the delay time, which can be described by an exact analytical formula. It is found that the delay time agrees with the phase time only for the case of strictly monochromatic waves.
We explore the dynamics of relativistic quantum waves in a potential step by using an exact solution to the Klein-Gordon equation with a point-source initial condition. We show that in both the propagation and Klein-tunneling regimes, the zitterbewegung effect manifests itself as a series of quantum beats of the particle density in the long-time limit. We demonstrate that the beating phenomenon is characterized by the zitterbewegung frequency and that the amplitude of these oscillations decays as ${t}^{\ensuremath{-}3/2}$. We show that the beating effect also manifests itself in the free Klein-Gordon and Dirac equations within a quantum shutter setup, which involve the dynamics of cutoff quantum states. We also find a time domain where the particle density of the point source is governed by the propagation of a main wavefront, exhibiting an oscillating pattern similar to the diffraction-in-time phenomenon observed in nonrelativistic systems. The relative positions of these wavefronts are used to investigate the time delay of quantum waves in the Klein-tunneling regime. We show that depending on the energy difference E\mathcal{E}E between the source and the potential step, the time delay can be positive, negative, or zero. The latter case corresponds to a super-Klein-tunneling configuration, where E\mathcal{E}E equals half the energy of the potential step.
Abstract We derive an exact analytical solution to the time-dependent Schrödinger equation based on a resonant state expansion, to explore the time-evolution of cutoff plane waves scattered by finite range potentials, within a shifted quantum shutter model. The latter allows to control both, the position of the shutter and an hypothetical detector, which are initially separated by a distance Δ X . The dynamical advance-time (negative delay-time), Δ T , is explored for the particular case of a delta potential well, by measuring the difference of the first maxima of the time-diffraction pattern of the probability density, corresponding to the free and delta potential cases. We show that in general, Δ T exhibits a monotonic behaviour as a function of Δ X , and derive a simple formula for the timescale by using the symmetry and rescaling properties of the dynamical solution. We demonstrate that below a critical value of the shutter-detector separation, Δ X c , the monotonic behaviour of Δ T is hindered due to an oscillatory phenomena of the probability density, known as persistent oscillations , that drastically distorts the maxima used in the measurement process. These persistent oscillations are periodic Rabi-type oscillations that arise from an interplay between the incidence energy of the initial quantum wave and the bound state of the system, and their frequency play an important role in the dynamics of the delay time.
Abstract We explore the dynamics of matter wave scattering in finite potential wells using analytical solutions of the Schrödinger equation within the framework of a quantum shutter model. We find that the incident wave interferes with the bound states of the quantum well, resulting in time-domain oscillations. These oscillations exhibit Rabi-type frequencies, characterised by the energy differences between the incident wave and the bound states of the quantum well. We show that in systems with double-bound states, the interference pattern is characterised by quantum beats in the time-dependent probability density. The period of these beatings depends on the energy difference between the bound states, which can be tuned by controlling the potential parameters. In the general case where bound, anti-bound, and resonant states coexist in the system spectrum, complex oscillations in the probability density arise from the interactions of the incident wave with different quantum states. We demonstrate that the bound states sector can effectively describe this complex behaviour, providing a simple and reliable analytical expression for the probability density in multilevel systems. This formula highlights the significant role of bound states, whose interaction with the incident wave dominates the transient probability density. This contrasts with conventional systems with potential barriers and wells, where resonances govern the wave dynamics.
Abstract We derive an exact analytic solution of time-dependent Schrödinger’s equation for cutoff quantum waves with exponential modulation to explore the dynamics along the internal region of a one-dimensional potential of arbitrary shape. The modulation of the wave packet is controlled by a single parameter given by its momentum distribution width, Δ, which allows to explore on the same footing the transient features of both cutoff spatially localized wave packets and extended plane waves. From the formal solution, we derive a simple and reliable formula that accurately describes both the buildup and decay of the probability density inside a resonant structure. The buildup-decay formula involves two exponential terms governed respectively by the resonance width Γ 1 of the lowest resonance of the system and the width Γ ¯ of a ‘virtual’ state induced by the incident wave packet. During the buildup process, we demonstrate that the contributions of both Γ 1 and Γ ¯ become essential. In the decay process, on the other hand, we find that there are two regimes characterized by a critical value Δ c , such that for Δ > Δ c the exponential decay is governed by Γ 1 , while for Δ < Δ c the exponential decay is governed by Γ ¯ . We also derive an analytic formula for a relevant timescale for the buildup-decay process, which depends in a simple fashion on the decay rates Γ 1 and Γ ¯ . Deviations from the exponential decay law at long times are also discussed.
We explore the dynamics of two-dimensional massless Dirac fermions within a quantum shutter approach, which involves the time evolution of an initial cutoff plane wave. We show that the probability density is governed by an interplay between diffraction in time and Zitterbewegung phenomena typical of relativistic quantum shutter systems with nonzero mass. The time diffraction appears as an oscillatory pattern in the probability density, similar to the effect predicted by Moshinsky in 1952 [Phys. Rev. 88, 625 (1952)] for Schr\"odinger free matter waves. The Zitterbewegung manifests itself as high-frequency oscillations embedded in the time-diffraction profile. We found that these two transient effects are induced by the transverse momentum component of the incident wave ky{k}_{y}ky​ that acts as an effective mass of the system. Furthermore, this effective mass can be manipulated by tuning the incidence angle of the initial quantum state, which allows us to control the frequencies of the transients. In particular, we demonstrate that near a normal incidence condition, the Zitterbewegung appears as a series of quantum beats in the probability density, with a beating frequency 2kyvF2{k}_{y}{v}_{F}2ky​vF​, where vF{v}_{F}vF​ is the Fermi velocity.
The analytical solution to the time-dependent Schr\"odinger equation for tunneling using cutoff plane-wave initial conditions is in general given by the sum of two types of terms that exhibit a transient behavior. The time evolution of the probability density for the $\ensuremath{\delta}$ potential is compared with the free case to investigate in this case the role of these transient terms for the delay time. We find, by a dynamical calculation, that the delay time arises from the interference between these transient terms and we show that at very long times it goes into the phase delay time, given by the energy derivative of the phase of the transmission amplitude.