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Dynamical Simulation of Electron Hoping in Double Quantum Rings

Chonlathep Kitsinthopchai, Sutee Boonchui

Published Jan 1, 2013• 0 citations
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Abstract

The wave packet of electron hoping through quantum rings under the static magnetic field is solved by transformation to the canonical form of certain system of differential equations. The Lorentz force leads to electron asymmetry which enhances the electron passing through a quantum ring while the Aharanov›Bohm effect (AB effect) reduces the probability of transmis- sion by phase shifted interference. For zero or similar magnetic field of both rings, the wave packet can pass both quantum rings to the exit quantum wire while different magnetic field of both rings prevent the second ringus injection of electron.

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ชลเทพ กิจสินธพชัย, สุธี บุญช่วย

แบบจำลองของอิเล็กตรอนที่เคลื่อนที่ผ่านวงแหวนควอนตัมภายใต้สนามแม่เหล็กคงตัวสามารถจำลองได้โดยการสร้างตัวแปลงเพื่อแก้ชุดสมการเชิงอนุพันธ์ให้ไปอยู่ในรูปคาโนนิคอลซึ่งแรงลอเรนซ์ที่เกิดขึ้นทำให้อิเล็กตรอนที่เคลื่อนผ่านวงแหวนมีการเอนเอียงไปด้านหนึ่ง และขณะเดียวกันผลจากปรากฏการณ์ของAharonov-Bohmซึ่งเป็นการเลื่อนเฟสได้ทำให้อิเล็กตรอนที่ปลายวงแหวนเกิดการแทรกสอดทำให้ความสามารถในการส่งผ่านลดลงผลลัพธ์จากการคำนวณแสดงให้เห็นว่าสาหรับกรณีที่ไม่มีสนามแม่เหล็กหรือสนามแม่เหล็กที่เกิดขึ้นบนวงแหวนทั้งสองมีค่าใกล้เคียงกันอิเล็กตรอนจะสามารถเคลื่อนที่ผ่านไปได้แต่สำหรับกรณีที่สนามแม่เหล็กมีค่าแตกต่างกันมากอิเล็กตรอนจะไม่สามารถเคลื่อนที่ผ่านวงแหวนวงแรกได้The wave packet of electron hoping through quantumrings under the static magnetic field is solved by transformation to thecanonical form of certain system of differential equations. The Lorentz forceleads to electron asymmetry which enhances the electron passing through aquantum ring while the Aharanov-Bohm effect (AB effect) reduces the probabilityof transmission by phase shifted interference. For zero or similar magneticfield of both rings, the wave packet can pass both quantum rings to the exitquantum wire while different magnetic field of both rings prevent the secondring’s injection of electron.

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The wave packet of electron hoping through quantum rings under the static magnetic field is solved by transformation to the canonical form of certain system of differential equations. The Lorentz force leads to electron asymmetry which enhances the electron passing through a quantum ring while the Aharanov›Bohm effect (AB effect) reduces the probability of transmis- sion by phase shifted interference. For zero or similar magnetic field of both rings, the wave packet can pass both quantum rings to the exit quantum wire while different magnetic field of both rings prevent the second ringus injection of electron.
แบบจำลองของอิเล็กตรอนที่เคลื่อนที่ผ่านวงแหวนควอนตัมภายใต้สนามแม่เหล็กคงตัวสามารถจำลองได้โดยการสร้างตัวแปลงเพื่อแก้ชุดสมการเชิงอนุพันธ์ให้ไปอยู่ในรูปคาโนนิคอลซึ่งแรงลอเรนซ์ที่เกิดขึ้นทำให้อิเล็กตรอนที่เคลื่อนผ่านวงแหวนมีการเอนเอียงไปด้านหนึ่ง และขณะเดียวกันผลจากปรากฏการณ์ของAharonov-Bohmซึ่งเป็นการเลื่อนเฟสได้ทำให้อิเล็กตรอนที่ปลายวงแหวนเกิดการแทรกสอดทำให้ความสามารถในการส่งผ่านลดลงผลลัพธ์จากการคำนวณแสดงให้เห็นว่าสาหรับกรณีที่ไม่มีสนามแม่เหล็กหรือสนามแม่เหล็กที่เกิดขึ้นบนวงแหวนทั้งสองมีค่าใกล้เคียงกันอิเล็กตรอนจะสามารถเคลื่อนที่ผ่านไปได้แต่สำหรับกรณีที่สนามแม่เหล็กมีค่าแตกต่างกันมากอิเล็กตรอนจะไม่สามารถเคลื่อนที่ผ่านวงแหวนวงแรกได้The wave packet of electron hoping through quantumrings under the static magnetic field is solved by transformation to thecanonical form of certain system of differential equations. The Lorentz forceleads to electron asymmetry which enhances the electron passing through aquantum ring while the Aharanov-Bohm effect (AB effect) reduces the probabilityof transmission by phase shifted interference. For zero or similar magneticfield of both rings, the wave packet can pass both quantum rings to the exitquantum wire while different magnetic field of both rings prevent the secondring’s injection of electron.
This article describes the Aharonov-Bohm (AB) effect of electron waves travelling in free space and its application to the observation of gauge fields (vector potentials). The AB effect is inconceivable in classical physics since an observable effect is produced on electrons passing through field-free spaces. Electrons can be affected only by Lorentz forces due to electromagnetic fields. The situation is different in quantum mechanics, where electrons show wave properties: the concept of force is no longer relevant, so electric field E and magnetic field B, defined as forces acting on a unit charge, take on secondary meanings. “Phase shifts” come into play, and the primary physical entities become neither E nor B but electrostatic potential V and vector potential A. These potentials interact with electron waves and shift their phases.
The traditional Aharonov Bohm effect (AB effect) attributes the light and dark fringes on screen to phase interference of electron matter wave. It presumes that the magnetic field outside the solenoid equals zero and particles receive no force. This system has long served as key support for wave particle duality. This paper completely abandons the hypothesis of wave interference and quantum phase. It establishes the model of electron vibrating point particle to reveal the real physical mechanism of AB effect. Fringes are not superposition product of waves. They are formed by differential deflection of charged electrons under magnetic force and Coulomb repulsion, followed by regular superposition of two upper and lower particle beams. Both magnetic force and Coulomb repulsion are long range forces. They only differ in effective action range and variation rule of force with distance. Comparative experiments using direct current solenoid and direct current long straight wire can falsify the unrealistic assumption that magnetic field outside coil cancels out to zero. Supporting single side shielding verification experiment can directly test the correctness of orthodox quantum interpretation. Electrons have no wave property. As long as point particles with inherent vibration are modulated by field, they can form fringe similar to interference fringe. Photons are also point particles with inherent vibration. When photons pass through slits, they interact with slit wall and produce deflection, so as to form photon diffraction and interference fringes. The hypothesis of wave particle duality is invalid.
We theoretically study the time-developed progress of resonant tunneling for an electron wave packet injected into a two-dimensional quantum ring (QR) by solving the time-dependent (TD) Schr\"odinger equation numerically. Focusing on an extraction of the angular momentum lz{l}_{z}lz​, we examine the TD features in the resonant tunneling electron by projection analysis in which the resulting TD wave function at the QR is decomposed into the (static) resonant states by calculating the inner products among them. This analysis reveals that the two-states approach is well applicable for the QR system and the cross terms between these two states are crucial for the TD vacillation of both the expectation value of the angular momentum $⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{l}}_{z}⟩$ and the electron density $\ensuremath{\rho}$. The quasidegeneracy of the resonant states causes a characteristic beating whose frequency is determined by the difference between the eigenenergies. We further study the corresponding TD phenomena under a magnetic field and find that the rotational direction in $\ensuremath{\rho}$ changes in accordance with the strength of the magnetic field. This feature seems to be very different from the classical prospect of the cyclotron motion where the application of the magnetic field determines the rotational direction uniquely.
Confined in a lossless conductive ring, fractional revival of electron wave packet can form a set of coherently coupled qubits. It is demonstrated that if several magnetic solenoids are mounted at specific locations around the outer rim of the ring, the local phase of electron wave can be dynamically adjusted to a desired value. This will change the wave packet revival evolution thereafter. In some senses, this phenomenon is similar to the Aharonov–Bohm effect. However, here no net magnetic flux passes through the ring which is the path loop of electron wave. By computer simulation, it is shown that some important qubit operations can be realized by this approach.
The AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux  through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc     , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.