Lecture Notes#

For an isolated system, the Schrödinger equation perfectly captures quantum evolution. But real systems interact with their surroundings — heat baths, photon fields, uncontrolled degrees of freedom. The Lindblad master equation describes how open systems evolve, accounting for both unitary dynamics and the irreversible effects of environmental coupling. It is the most general form of Markovian (memoryless) evolution compatible with quantum mechanics.

Homework#

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

(a) Show that \(\text{Tr}(\mathrm{d}\hat{\rho}/\mathrm{d}t) = 0\) (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. 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 \(\hat{\rho}_{01}(0) eq 0\), show that \(\hat{\rho}_{01}(t) = \hat{\rho}_{01}(0)\,\mathrm{e}^{-t/(2T_1)}\). Coherences decay at half the relaxation rate.

3. A qubit undergoes pure dephasing via \(L = \sqrt{\Gamma_\varphi}\hat{\sigma}^z\). 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. 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 \(\hat{\rho}_\infty \propto \mathrm{e}^{-\beta\hbar\omega\hat{\sigma}^z/2}\) (Boltzmann distribution).

5. In the quantum jump picture, the system evolves under \(H_\text{eff} = -\frac{\mathrm{i}\hbar\Gamma}{2}\vert 1\rangle\langle 1\vert\) (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. 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 \(\mathcal{P}(t) = \text{Tr}(\hat{\rho}(t)^2)\). Show that \(\frac{\mathrm{d}\mathcal{P}}{\mathrm{d}t} = -2\Gamma(\mathcal{P} - 1/2)\). Solve to get \(\mathcal{P}(t) = \frac{1}{2} + \frac{1}{2}\mathrm{e}^{-2\Gamma t}\).

(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. A Lindblad equation has two operators: decay \(L_1 = \sqrt{\Gamma_1}\hat{\sigma}_-\) and dephasing \(L_2 = \sqrt{\Gamma_\varphi}\hat{\sigma}^z\).

(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?