A mathematical aspect of a tunnel-junction for spintronic qubit

Abstract

We consider the Dirac particle living in the 1-dimensional configuration space with a junction for a spintronic qubit. We give concrete formulae explicitly showing the one-to-one correspondence between every self-adjoint extension of the minimal Dirac operator and the boundary condition of the wave functions of the Dirac particle. We then show that the boundary conditions are classified into two types: one of them is characterized by two parameters and the other is by three parameters. Then, we show that Benvegnù and Dabrowski's four-parameter family can actually be characterized by three parameters, concerned with the reflection, penetration, and phase factor.Tunnel-Junction Formulae for Spintronic Qubit 2 as a black box so that it has mathematical arbitrariness. We regard the wires as the union of the intervals, (−∞, −Λ) ∪ (Λ, ∞), Λ ≥ 0, for mathematical simplicity. Namely, the segment [−Λ, Λ] with length 2Λ plays a role of the junction. Many physicists have investigated the individual boundary condition of the wave functions of the Dirac particle for the corresponding self-adjoint extension of the Dirac operator in the case of the point interaction (i.e., Λ = 0) [1,2,8,14,30]. Meanwhile, the Dirac operator consists of the combination of the Dirac matrices and the momentum operator of electron. The boundary conditions of the self-adjoint extensions of the momentum operators have been studied by mathematicians [18,26]. We note that there is a general theory in mathematics, called boundary triple, to handle the boundary condition, and the theory has still been developed [6,7,11,10,13]. In Refs. [15,31] the appearance of a phase factor was proved for the Schrödinger particle under the same mathematical set-up as ours. For our Dirac particle, in the case where Λ = 0, Benvegnù and Dabrowski showed that a phase factor appears in their four-parameter family (see Eq.(15) of Ref. [8]). In addition, they showed how the boundary condition affects the spin. The description of boundary condition by the individual one-parameter families have been studied in Refs. [1,14,20,30] (also see Eqs. (17) and (18) of Ref.[8]). We will go ahead and make an in-depth research for the Dirac particle. Thus, we employ the minimal Dirac operator for the Hamiltonian. In this paper, we follow the machinery in Refs. [8,15,18,26,31] based on the von Neumann's theory [27,34], in which all the self-adjoint extensions of our minimal Dirac operator are parameterized by U ∈ U(2), where U(2) is unitary group of degree 2.We prove that all the boundary conditions of wave functions of our Dirac particle are completely classified into the two types (See Corollary 4.5). One of them is the type that states the wave functions do not pass through the junction, and described by two parameters, γ L , γ R ∈ C with |γ L | = |γ R | = 1, concerned with the reflections at −Λ and at +Λ, respectively. The other is the type that states the wave functions do pass through the junction, and described by Benvegnù and Dabrowski's four-parameter family (see Theorem 4.2, Theo...

One-Dimensional Tunnel-Junction Formula for the Schrödinger Particle

“…The following lemma says that U ∈ U (2) can be decomposed into the product of an element of U (1) and an element of SH. Although this lemma was already proved in [3,Proposition 4.3], we here give a simpler proof than that of [3, Proposition 4.3]:…”

“…We will propose a mathematical idea to make a qubit from a Schrödinger particle through the tunnel-junction formula by controlling the phase factor, even though the spin of the Schrödinger particle cannot be used. In the case where the electron spin should be considered, we studied similar problem for a single relativistic electron as the Dirac particle [3].…”

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

“…For more details on self-adjoint extensions of the Dirac operator on metric graphs we refer the reader to [17,46]. We also mention [35], where boundary conditions for 1-D Dirac operators are studied for a model of quantum wires. Definition 2.3.…”