Technical Summary of My Bachelor's Thesis
(A) Majorana Zero Modes in Topological Superconductors
Majorana Zero Modes are zero-energy excitations that emerge at the boundaries of topological superconductors. These modes are described by the Bogoliubov-de Gennes equation, and their existence is tied to the breaking of electron-hole symmetry in the presence of a strong magnetic field. The Hamiltonian for a one-dimensional p-wave superconductor is given by:
\[ H = -\mu \sum_x c_x^\dagger c_x - \frac{1}{2} \sum_x \left( t c_x^\dagger c_{x+1} + \Delta e^{i\phi} c_x c_{x+1} + h.c. \right) \]where:
- \(\mu\) is the chemical potential,
- \(t\) is the hopping amplitude,
- \(\Delta\) is the superconducting pairing amplitude,
- \(\phi\) is the superconducting phase.
When \(\mu = 0\) and \(t = \Delta\), the system hosts Majorana zero modes at its ends, described by the operators \(\gamma_1\) and \(\gamma_2\). These operators satisfy the anti-commutation relations:
\[ \{\gamma_i, \gamma_j\} = 2\delta_{ij}, \quad \gamma_i^2 = 1 \]These Majorana modes are non-local, therefore the quantum information encoded in them is distributed across the system, providing topological protection against local perturbations. The non-locality arises because the Majorana operators \(\gamma_1\) and \(\gamma_2\) are spatially separated, and any local noise affecting one mode will not easily corrupt the encoded quantum information.
The Majorana zero modes can be understood as fractionalized fermions**. In a p-wave superconductor, the fermionic operators \(c_x\) and \(c_x^\dagger\) can be expressed in terms of Majorana operators:
\[ c_x = \frac{e^{-i\phi/2}}{2} (\gamma_{B,x} + i\gamma_{A,x}), \quad c_x^\dagger = \frac{e^{i\phi/2}}{2} (\gamma_{B,x} - i\gamma_{A,x}) \]where \(\gamma_{A,x}\) and \(\gamma_{B,x}\) are Majorana operators at site \(x\). When \(\mu = 0\) and \(t = \Delta\), the Hamiltonian simplifies to:
\[ H = -i\frac{t}{2} \sum_{x=1}^{N-1} \gamma_{B,x} \gamma_{A,x+1} \]In this regime, the Majorana modes at the ends of the chain, \(\gamma_1 = \gamma_{A,1}\) and \(\gamma_2 = \gamma_{B,N}\), are decoupled from the rest of the system, forming a pair of zero-energy modes. These modes can be combined to form a non-local fermion:
\[ f = \frac{1}{2} (\gamma_1 + i\gamma_2) \]This fermion operator \(f\) and its conjugate \(f^\dagger\) can be used to encode a qubit, where the quantum information is stored in the parity of the fermionic state.
(B) Quantum Dots and Majorana Fermion Manipulation
To manipulate Majorana fermions in practice, we can couple them to quantum dots, which are nanoscale semiconductor structures that can trap electrons. The coupling between a quantum dot and a Majorana fermion can be described by the Hamiltonian:
\[ H_T = \sum_i \left( v_i d \gamma_i - v_i^* d^\dagger \gamma_i \right) \]where:
- \(d\) and \(d^\dagger\) are the annihilation and creation operators for the quantum dot,
- \(v_i\) is the coupling strength between the quantum dot and the Majorana fermion \(\gamma_i\).
By tuning the coupling strengths \(v_i\), we can control the state of the Majorana fermions and perform quantum operations. For example, consider a system where two Majorana fermions \(\gamma_1\) and \(\gamma_2\) are coupled to a quantum dot. The effective Hamiltonian in this case is:
\[ H_{12} = \epsilon d^\dagger d + (v_1^* d^\dagger - v_1 d) \gamma_1 + (v_2^* d^\dagger - v_2 d) \gamma_2 \]where \(\epsilon\) is the energy of the quantum dot. The coupling terms \((v_1^* d^\dagger - v_1 d) \gamma_1\) and \((v_2^* d^\dagger - v_2 d) \gamma_2\) describe the interaction between the quantum dot and the Majorana fermions.
The Hamiltonian \(H_{12}\) can be diagonalized to find the eigenstates and eigenvalues. In the basis \(\{|0\rangle_D |0\rangle_{M12}, |1\rangle_D |1\rangle_{M12}, |0\rangle_D |1\rangle_{M12}, |1\rangle_D |0\rangle_{M12}\}\), the Hamiltonian takes the form:
\[ H_{12} = \begin{bmatrix} 0 & v_1 - iv_2 & 0 & 0 \\ v_1 + iv_2 & \epsilon & 0 & 0 \\ 0 & 0 & 0 & v_1 + iv_2 \\ 0 & 0 & v_1 - iv_2 & \epsilon \end{bmatrix} \]The eigenvalues of this Hamiltonian are given by:
\[ E = \epsilon - \sqrt{\left(\frac{\epsilon}{2}\right)^2 + |v_1|^2 + |v_2|^2 \mp 2|v_1 v_2| \sin(\varphi_1/2)} \]where \(\varphi_1 = 2\arg(v_1/v_2)\). To ensure degeneracy in the energy levels, we set \(\sin(\varphi_1/2) = 0\), which implies \(\varphi_1 = 2n\pi\). This condition can be achieved by tuning the magnetic flux \(\Phi_1\) through the system, where \(\varphi_1 = \Phi_1 / \Phi_0\) and \(\Phi_0 = h/2e\).
When the degeneracy condition is met, the Hamiltonian simplifies, and we can define a new effective Majorana operator:
\[ \gamma_{12} = \frac{1}{v} (|v_1| \gamma_1 + |v_2| \gamma_2) = u \gamma_1 + v \gamma_2 \]where \(v = \sqrt{|v_1|^2 + |v_2|^2}\). This new operator \(\gamma_{12}\) allows us to rewrite the Hamiltonian in a simpler form:
\[ H_T = v (d^\dagger + d) \gamma_{12} \]By controlling the parameters \(v_1\) and \(v_2\), we can perform operations on the Majorana qubit, such as rotations in the **Bloch sphere** representation. For example, the operator \(\gamma_{12}\) corresponds to a rotation by \(\pi\) around an axis in the \(xy\)-plane:
\[ \gamma_{12} = u \sigma_x + v \sigma_y \]where \(\sigma_x\) and \(\sigma_y\) are Pauli matrices. This rotation can be used to implement quantum gates, which opens up the pathway to quantum computing.