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arXiv:1511.00608 · Journal of Statistical Mechanics Theory and Experiment

Quantum noise with exchange and tunnelling: predictions for a two-particle scattering experiment with time-dependent oscillatory potentials

Enrique Colomés, Damiano Marian, Xavier Oriols

Published May 1, 2016• 1 citation
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Abstract

Abstract. Quantum noise with exchange and tunnelling is studied within time-dependent wave packets. A novel expression for the quantum noise of two identical particles injected simultaneously from opposite sides of a tunnelling barrier is presented. Such quantum noise expression provides a physical (non-spurious) explanation for the experimental detection of two electrons at the same side under static potentials. Numerical simulations of the two-particle scattering probabilities in a double barrier potential with an oscillatory well are performed. The dependence of the quantum noise on the electron energy and oscillatory frequency is analysed. The peculiar behaviour of the dependence of the quantum noise on such parameters is proposed as a test about the soundness of this novel quantum noise expression, for either static or oscillatory potentials.

DOI
10.1088/1742-5468/2016/05/054009

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Abstract. Quantum noise with exchange and tunnelling is studied within time-dependent wave packets. A novel expression for the quantum noise of two identical particles injected simultaneously from opposite sides of a tunnelling barrier is presented. Such quantum noise expression provides a physical (non-spurious) explanation for the experimental detection of two electrons at the same side under static potentials. Numerical simulations of the two-particle scattering probabilities in a double barrier potential with an oscillatory well are performed. The dependence of the quantum noise on the electron energy and oscillatory frequency is analysed. The peculiar behaviour of the dependence of the quantum noise on such parameters is proposed as a test about the soundness of this novel quantum noise expression, for either static or oscillatory potentials.
Two-particle scattering probabilities in tunneling scenarios with exchange interaction are analyzed with quasi-particle wave packets. Two initial one-particle wave packets (with opposite central momentums) are spatially localized at each side of a barrier. After impinging upon a tunneling barrier, each wave packet splits into transmitted and reflected components. When the initial two-particle anti-symmetrical state is defined as a Slater determinant of any type of (normalizable) one-particle wave packet, it is shown that the probability of detecting two (identically injected) electrons at the same side of the barrier is different from zero in very common (single or double barrier) scenarios. In some particular scenarios, the transmitted and reflected components become orthogonal and the mentioned probabilities reproduce those values associated to distinguishable particles. These unexpected non-zero probabilities are still present when non-separable Coulomb interaction or non-symmetrical potentials are considered. On the other hand, for initial wave packets close to Hamiltonian eigenstates, the usual zero two-particle probability for electrons at the same side of the barrier found in the literature is recovered. The generalization to many-particle scattering probabilities with quasi-particle wave packets for low and high phase-space density are also analyzed. The far-reaching consequences of these non-zero probabilities in the accurate evaluation of quantum noise in mesoscopic systems are briefly indicated.
Quantum transport is commonly studied with the use of quasi-particle infinite- extended states. This leads to a powerful formalism, the scattering-states theory, able to capture in compact formulas quantities of interest, such as average current, noise, etc.. However, when investigating the spatial size-dependence of quasi-particle wave packets in quantum noise with exchange and tunneling, unexpected new terms appear in the quantum noise expression. For this purpose, the two particle transmission and reflection probabilities for two initial one-particle wave packets (with opposite central momentums) spatially localized at each side of a potential barrier are studied. After the interaction, each wave packet splits into a transmitted and a reflected component. It can be shown that the probability of detecting two (identically injected) electrons at the same side of the barrier is different from zero in very common (single or double barrier) scenarios. This originates an increase of quantum noise which cannot be obtained through the scattering states formalism.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.
Quantum tunneling, a quintessential quantum phenomenon, challenges classical intuitions by allowing particles to pass through seemingly insurmountable barriers. This study delves into the intricate dynamics of electron tunneling, focusing on the critical role of sub-barrier interactions and complex time processes. Through a combination of experimental measurements and simulations based on Feynman’s path integral of complex time trajectories, we reveal that the sub-barrier potential interaction introduces an asymmetric electron wave packet probability distribution along the real time axis and a marked delay at the tunneling exit. The energy-dependent behavior of this delay in electronic tunneling agrees with that observed in the previous atomic tunneling studies, suggesting a universal mechanism underlying quantum tunneling across different scales. Our findings underscore the necessity of incorporating both the wave and particle aspects of quantum entities to fully comprehend tunneling dynamics, which holds promise for an intuitive interpretation of complex-number quantum processes.
In this paper, the tunnelling of a particle through a potential barrier is investigated in the presence of a time-dependent perturbation. The latter is attributed to the process of the energy measurement of the scattered particle. The time-dependent Schrödinger equation of the model is exactly solved. The probability density inside the barrier is calculated from the obtained wave function, proving that the tunnelling dynamics is determined not only by the transmitted and reflected waves but also by their interference. Furthermore, the interference term is time-dependent and contribute to the scattering process duration. The tunnelling time is calculated as the time needed to get the probability density inside the barrier to zero. This is the minimum duration of the measurement process before detecting the particle beyond the barrier. Based on this, a new method of estimating the tunnelling time by energy experimental measuring is proposed.