Lecture Notes#

Homework#

1. Amplitude Damping Lindblad. Consider a qubit under amplitude damping, \(L = \sqrt{\Gamma}\hat{\sigma}_-\) where \(\hat{\sigma}_- = \vert 0\rangle\langle 1\vert\).

(a) Show that \(\operatorname{Tr}(\mathrm{d}\hat{\rho}/\mathrm{d}t) = 0\) (trace is conserved). Use the cyclic property of trace.

(b) Verify \(\hat{\sigma}_+\hat{\sigma}_- = \vert 1\rangle\langle 1\vert\) (projector onto excited state). Use this to write the loss term explicitly.

(c) Show that if \(\hat{\rho}(0)\) is Hermitian, then \(\mathrm{d}\hat{\rho}/\mathrm{d}t\) is also Hermitian (Hermiticity is preserved).

2. Decay Rate Dynamics. A qubit decays via amplitude damping at rate \(\Gamma_1 = 1/T_1\). Start with \(\hat{\rho}(0) = \vert 1\rangle\langle 1\vert\).

(a) Write the Lindblad master equation explicitly for populations \(\hat{\rho}_{00}(t)\) and \(\hat{\rho}_{11}(t)\).

(b) Solve to find \(\hat{\rho}_{11}(t) = \mathrm{e}^{-t/T_1}\) and \(\hat{\rho}_{00}(t) = 1 - \mathrm{e}^{-t/T_1}\).

(c) Show the steady state is \(\hat{\rho}_\infty = \vert 0\rangle\langle 0\vert\) (all population in the ground state).

(d) Starting from \(\hat{\rho}(0)\) with coherence \(\hat{\rho}_{01}(0) \neq 0\), show that \(\hat{\rho}_{01}(t) = \hat{\rho}_{01}(0)\,\mathrm{e}^{-t/(2T_1)}\). Coherences decay at half the relaxation rate.

3. Pure Dephasing Lindblad. A qubit undergoes pure dephasing via \(L = \sqrt{\Gamma_\varphi}\hat{Z}\). Start with \(\hat{\rho}(0) = \vert+\rangle\langle+\vert\).

(a) Write \(\hat{\rho}(0)\) explicitly as a \(2\times 2\) matrix.

(b) Apply the Lindblad superoperator: show that diagonal elements (populations) are unchanged and off-diagonal elements decay as \(\hat{\rho}_{01}(t) = \mathrm{e}^{-2\Gamma_\varphi t}\hat{\rho}_{01}(0)\).

(c) As \(t \to \infty\), show that \(\hat{\rho}_\infty = I/2\) (maximally mixed state). Explain physically: why does dephasing destroy coherence while preserving energy?

4. Thermal Radiation Coupling. An atom couples to a thermal radiation field at temperature \(T\). The relevant processes are spontaneous emission at rate \(A\), absorption at rate \(Bn_\text{th}\) (where \(n_\text{th} = 1/(\mathrm{e}^{\hbar\omega/k_BT}-1)\)), and dephasing at rate \(\gamma_\varphi\).

(a) Write the three Lindblad operators \(L_1\), \(L_2\), \(L_3\) for these processes.

(b) At \(T = 0\): \(n_\text{th} \to 0\), so only spontaneous emission survives. Write the simplified master equation and find the steady state.

(c) At high temperature (\(k_BT \gg \hbar\omega\)): \(n_\text{th} \approx k_BT/\hbar\omega\). Show that the steady state approaches \(\hat{\rho}_\infty \propto \mathrm{e}^{-\beta\hbar\omega\hat{Z}/2}\) (Boltzmann distribution).

5. Quantum Problem. In the quantum jump picture, the system evolves under \(\hat{H}_\text{eff} = -\frac{\mathrm{i}\hbar\Gamma}{2}\vert 1\rangle\langle 1\vert\) (for amplitude damping) between jumps.

(a) For initial state \(\vert\psi(0)\rangle = \vert 1\rangle\), show that the no-jump evolution gives \(\vert\psi(t)\rangle \propto \mathrm{e}^{-\Gamma t/2}\vert 1\rangle\). The squared norm \(\Vert\psi(t)\Vert^2 = \mathrm{e}^{-\Gamma t}\) is the probability that no jump has occurred.

(b) If a jump occurs at time \(t\), the state becomes \(\hat{\sigma}_-\vert\psi(t)\rangle \propto \vert 0\rangle\) (ground state). Interpret this physically: what is the jump event?

(c) Show that averaging over all possible trajectories (no jump with probability \(\mathrm{e}^{-\Gamma t}\), one jump at time \(\tau\) with probability \(\Gamma\mathrm{e}^{-\Gamma\tau}\mathrm{d}\tau\)) reproduces the Lindblad master equation solution \(\hat{\rho}(t) = \mathrm{e}^{-\Gamma t}\vert 1\rangle\langle 1\vert + (1 - \mathrm{e}^{-\Gamma t})\vert 0\rangle\langle 0\vert\).

6. Depolarizing Lindblad Equation. A qubit undergoes depolarizing noise via \(L_\mu = \sqrt{\Gamma/3}\hat{\sigma}^\mu\) for \(\mu = x,y,z\).

(a) Show that \(\hat{\rho}_\infty = I/2\) is a steady state: \(\mathcal{L}(I/2) = 0\). (Hint: Use \(\hat{\sigma}^\mu(I/2)\hat{\sigma}^\mu = I/2\) for each Pauli.)

(b) Starting from \(\hat{\rho}(0) = \vert 0\rangle\langle 0\vert\), define purity \(\mathcal{P}(t) = \operatorname{Tr}(\hat{\rho}(t)^{2})\). Using the identity \(\sum_{\mu}\hat{\sigma}^{\mu}\hat{\rho}\hat{\sigma}^{\mu} = 2\hat{I} - \hat{\rho}\) for any qubit state, show that the Bloch vector decays as \(\dot{\boldsymbol{r}} = -(4\Gamma/3)\boldsymbol{r}\), hence \(\dot{\mathcal{P}} = -(8\Gamma/3)(\mathcal{P} - 1/2)\). Solve to get \(\mathcal{P}(t) = \tfrac{1}{2} + \tfrac{1}{2}\mathrm{e}^{-8\Gamma t/3}\).

(c) Identify the Kraus operators \(\{K_\mu\}\) for the depolarizing channel (integrated Lindblad evolution for short time \(\mathrm{d}t\)). Show the channel matches the form from §6.3.3.

7. Multiple Jump Operators. A Lindblad equation has two operators: decay \(L_1 = \sqrt{\Gamma_1}\hat{\sigma}_-\) and dephasing \(L_2 = \sqrt{\Gamma_\varphi}\hat{Z}\).

(a) Write the full master equation for \(\dot{\hat{\rho}}_{11}\) and \(\dot{\hat{\rho}}_{01}\). Show that populations decay at rate \(\Gamma_1\) and coherences decay at rate \(\Gamma_1/2 + 2\Gamma_\varphi\).

(b) Identify the dephasing time \(T_2\) from the coherence decay rate. Show that \(\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\varphi}\) where \(T_1 = 1/\Gamma_1\) and \(T_\varphi = 1/(2\Gamma_\varphi)\).

(c) Explain physically why \(T_2 \leq 2T_1\). Under what condition is this bound saturated?