Based on the stationary phase concept and the effective one-band Hamil-tonian with the Dresselhaus spin-orbit coupling, we present the numerical results of the tunneling time through a realistic InGaAs/InAlAs/InGaAs resonant symmetric structure. It is shown that the polarization efficiency of the structure has a well-defined resonance behavior, which leads to a consid-erable spin polarization of electrons tunneling through. In the lower energy region, the ratio between the tunneling times of electrons with opposite spin orientation can vary over a few orders in magnitude. The results indicate that the Dresselhaus spin-orbit coupling separates the time-dependent re-sponse of differently spin-polarized tunneling electrons. Further, the large and tunable ratio of the tunneling times provides a possible way to con-struct a dynamic spin filter. The characteristic time of such devices also has been estimated and presented, showing simple functional dependencies on the barrier thickness and the well width. The dependencies can be exploited to design spintronic devices working in the desired frequencies.
Chapter 3
Aharonov-Bohm quantum gates with ballistic electrons
The physical implementation of a quantum computer continues to pose a great challenge. Among the numerous schemes to implement quantum com-puters, solid state micro- and nano- systems draw special attention because of their obvious advantages: scalability, miniaturizability and flexibility in design. Quantum computation using ballistic electrons has been proposed recently as an attractive candidate [36, 37, 38]. In this scheme we use bal-listic electrons as flying qubits in one dimensional quantum wires within the dual rail representation [38, 44]. That is, two adjacent 1D quantum wires, called the 0- and the 1-rail respectively, are used as the physical implemen-tation of a single qubit. The logic state |0i is defined as the presence of a single electron in the 0-rail while the logic state |1i as in the 1-rail. Initial-ization and measurement of the qubit states are done by coupling each qubit rail to a single electron transistor, as was proposed in [38]. The construction of quantum gates deserves more discussion so from hereon we pay special attention to it.
After the first idea of utilizing quantum interference effects to perform logic functions has been proposed by Datta et el. [40], the property of quantum waveguide and quantum network has been widely investigated.
Among them are quantum interference transistors [40, 47], serial stubs [30]
and Ahanorov-Bohm rings with multiple arms [31], to list only a few. A complete set of logic functional devices has also been proposed by Wu. et el. [45], leading to the possibility of performing massive parallel computing by electron wave. Those previous works, however, focus mainly on the ma-nipulation of transmission probability but not on the accompanying changes on the phase of the wave. That is, they concentrated on the possibility to perform classical computation but miss the even more natural application to perform quantum computation. Thus we are motivated to explore the ca-pacity of quantum interference devices to perform single-qubit operation in the context of quantum computation using ballistic electrons. But to achieve that end we would like to look at the existing quantum network theory in a new setting first.
3.1 A detour to microwave engineering
In this detour we will first provide an equivalent description of the quantum network, which brings us more flexibilities in designing the quantum network and gives insight into the capacity of quantum interference devices. Then we adopt those ideas to create a single-qubit quantum gate for ballistic electrons.
All knowledge about microwave engineering used in this section can be found in [34].
It is well-known that both the time-independent Schr¨odinger equation and the the source-free Maxwell equations take the form of the Helmholtz
equations. Further resemblance appears when we consider the TEM solutions of the Maxwell equations in waveguides, i. e., the transmission line equation [34], and the one-dimensional Schr¨odinger equation in constant potential.
The analogy can be most directly seen as follows: In the transmission line theory we concern only two integrated physical quantities of the electric circuit, the electrical potential V (x) and the electrical current I(x), but not the detailed field solutions of the Maxwell equations. Their solutions in a lossless transmission line are:
where Z0 is the characteristic impedance of the transmission line and β is the propagation constant. However, in constant potential the time-independent one-dimensional Schr¨odinger equation has solutions:
ψ(x) = a1eikx+ a2e−ikx, conductance in the quantum network as Z10 in the transmission line, those two sets of solutions take the same form. Their boundary conditions are also the same. In the transmission line equation we require the continuity of the voltage V (x) and the conservation of the current I(x) at each intersec-tion; in a one-dimensional quantum network we require the continuity of the wave function ψ(x) and the conservation of the current ∂ψ(x)∂x by the Griffiths boundary conditions [35, 47]. Thus we can draw freely the already
sophisti-cated circuits and techniques from the microwave engineering to apply to a quantum network.
Then we immediately see that a quantum interference transistor in [47]
amounts to a single stub. Total transmission occurs (see Fig.7 in that refer-ence) when kL/π = n2, n ∈ Z, or L = nλ4 , i.e., when the single stub acts as a quarter-wave transformer such that the infinite load is transformed into a zero load. An impurity introduced into the quantum network, modeled as a δ-function in [33], is like a conductance connected in series. It can be put at the intersection of multiple segments of 1D quantum wires to match the wave impendence of the quantum network. The inclusion of external magnetic field breaks the reciprocity of the quantum network, creating anisotropic quantum interference devices. These devices include the isolator and the quantum cir-culator, both of which prevent reflected waves of the next stage from further interfering with the input waves. All the above mentioned structures give us more flexibilities in designing quantum network to do computation.
Having seen a variety of quantum interference devices, we now want to make use of them to perform single-qubit operation. Again let’s look at some of the existing microwave circuits first. The (180◦) hybrid works as a Hadmard gate. The quadrature(90◦) hybrid (see the figure below) is espe-cially interesting. Let the input and output electron have a wave number
k =√
2m∗E/~, where E stands for the electron energy and m∗ for the elec-tron effective mass. Also, let ka and kb be the elecelec-tron wave number in the corresponding segments of 1D quantum wires in Fig. 3.1. The length of each
of the four internal segments of 1D quantum wires is chosen to be λ4 with respect to the corresponding wave number in that segment. Connecting four phase shifters [38] to it, the transfer matrix of this quadrature(90◦) hybrid
would be can be constructed to act as arbitrary single-qubit quantum gate.
There are two problems with this construction, however. To have an identity or a σxgate, α has to approach to infinity or zero. This is impractical for α is related to the voltage applied to different segments of 1D quantum wires. The second problem arises since we require the length of each of the four internal segments of 1D quantum wires to be λ4 with respect to the corresponding wave number in that segment. Different ka and kb require different length for the corresponding segments. If we have no means to vary the length of each segment of quantum wire, then the functionality of such quantum gates could not be changed after fabrication.
The ability to construct quantum gates by total electrical means is not all satisfying. Adding an external magnetic field to electric circuits, however, can further modulate the phases of the electronic waves of the ballistic electrons (the Aharonov-Bohm (AB) effect [39, 40]). So, we propose in this chapter a system of one-dimensional (1D) quantum wires incorporating an array of nano-rings and nano-sized magnets [43] which can act as a new architecture to perform quantum computation. Each quantum gate in this architecture is controlled dynamically by flipping the magnetization of the nano-sized magnets and changing the chemical potential for the ballistic electrons. This provides the opportunity to program dynamically a quantum computer the
same way as we do a classical one.