The Exercise

A qubit, an atom with a ground state $|0\rangle$ and an excited state $|1\rangle$ , loses its excitation to its surroundings at a rate $\lambda$ . Its density operator obeys a master equation in Lindblad form whose one Lindblad operator is the lowering operator, which takes $|1\rangle$ to $|0\rangle$ , scaled by $\sqrt{\lambda}$ . The exercise is to solve this equation from an initial state $\rho_0$ and to show that the state at time $t$ is the image of $\rho_0$ under the amplitude damping channel with damping parameter $\gamma = 1 - e^{-\lambda t}$ .

The Master Equation

The lowering operator $\sigma_-$ of the qubit is the operator on its state space that takes $|1\rangle$ to $|0\rangle$ and $|0\rangle$ to zero, $\sigma_- = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = |0\rangle\langle 1| .$ (1)^✦ The raising operator $\sigma_+$ of the qubit is the operator on its state space that takes $|0\rangle$ to $|1\rangle$ and $|1\rangle$ to zero, $\sigma_+ = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = |1\rangle\langle 0| .$ (2)^✦

The Lindblad generator $\mathcal{L}_\lambda$ of amplitude damping at a real rate $\lambda$ is the map that takes an operator $\rho$ on the qubit to $\mathcal{L}_\lambda(\rho) = -\frac{\lambda}{2}\Bigl(\sigma_+\sigma_-\,\rho + \rho\,\sigma_+\sigma_- - 2\,\sigma_-\,\rho\,\sigma_+\Bigr),$ (3)^✦ where $\sigma_+\sigma_- = |1\rangle\langle 1|$ is the projector onto the excited state. The master equation of the exercise is $d\rho/dt = \mathcal{L}_\lambda(\rho)$ .

The Solution

For an operator $\rho_0$ on the qubit, write $\rho_{ij} = \langle i|\rho_0|j\rangle$ for its matrix elements. The damped state $\rho(t)$ of the qubit at time $t$ , for a rate $\lambda$ and an initial operator $\rho_0$ , is the operator $\rho(t) = \begin{pmatrix} \rho_{00} + \bigl(1 - e^{-\lambda t}\bigr)\,\rho_{11} & e^{-\lambda t/2}\,\rho_{01} \\ e^{-\lambda t/2}\,\rho_{10} & e^{-\lambda t}\,\rho_{11} \end{pmatrix}.$ (4)^✦

The operation elements $E_0(t)$ and $E_1(t)$ of amplitude damping at rate $\lambda$ after a time $t$ are the operators on the qubit $E_0(t) = \begin{pmatrix} 1 & 0 \\ 0 & e^{-\lambda t/2} \end{pmatrix}$ (5) and $E_1(t) = \begin{pmatrix} 0 & \sqrt{1 - e^{-\lambda t}} \\ 0 & 0 \end{pmatrix}.$ (6)

The Claim

Theorem 1. Let $\lambda$ be a real rate, $\rho_0$ any operator on the qubit, and $\rho(t)$ the damped state (4). Then $\rho(t)$ starts at $\rho_0$ , $\rho(0) = \rho_0 ,$ (7)^✦ and at every real time $t$ it satisfies the master equation with the generator (3), $\frac{d\rho}{dt}(t) = \mathcal{L}_\lambda\bigl(\rho(t)\bigr).$ (8)^✦ If moreover $\lambda \ge 0$ and $t \ge 0$ , then $\rho(t)$ is the image of $\rho_0$ under the channel with operation elements (5) and (6), $\rho(t) = E_0(t)\,\rho_0\,E_0(t)^\dagger + E_1(t)\,\rho_0\,E_1(t)^\dagger .$ (9)^✦

What Is Checked

The claim is stated over the damped state (4), the Lindblad generator (3) built from the lowering and raising operators (1) and (2), and the operation elements (5) and (6).

About these constructions the library proves the three parts of the theorem: the initial value (7), the master equation (8) and the operator sum (9).