Chapter 10

Open systems and decoherence

Why the world looks classical: density matrices, CPTP maps, Lindblad.

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10.1Why we need the density matrixNo lab owns an isolated system: qubits soak in their environment, atoms fly out of an oven at random. The tool for "not knowing" and "having no state of one's own" is the density matrix — and this chapter sets it in motion.
10.2Mixed states and the shrinking Bloch sphereEvery state of a qubit lives inside one unit ball: pure states on the surface, mixed states in the interior, and "complete ignorance" at the centre. Unitary evolution is rotation; the environment's damage is contraction.
10.3Quantum operations and CPTP mapsTurn the sentence "system and environment evolve unitarily together, then look only at the system" into mathematics, and out come Kraus operators and CPTP maps — the most general legitimate evolution of a quantum state.
10.4The Lindblad master equationSlice a quantum channel into infinitesimal slabs of time and you get the equation of motion for open systems. Use it to solve the damped two-level atom in full, and read off T₁, T₂ and the famous inequality T₂ ≤ 2T₁.
10.5The physical picture of decoherenceThe environment is an uninvited measurer: every scattered photon takes a "photograph" of the system. Coherences get erased at an exponential rate, and for macroscopic objects the rate is absurdly fast — this is why Schrödinger's cat is never seen.
10.6A first look at quantum DarwinismYou have never directly seen anything — what you see are record copies in the environment. Classical objectivity = one piece of information copied an astronomical number of times across the environment, and only pointer-state information can be copied that way.

Start with 10.1 Why we need the density matrix →

Formula reference

Basic equations

Time-dependent Schrödinger equationsource
$\ii\hbar\frac{\partial}{\partial t}\Psi(x,t)=\hat H\,\Psi(x,t)$
The equation of motion of quantum mechanics; first order in time
Time-independent Schrödinger equationsource
$-\frac{\hbar^2}{2m}\frac{\dd^2\psi}{\dd x^2}+V(x)\psi=E\psi$
An eigenvalue problem; solve for E and ψ
Hamiltonian operatorsource
$\hat H=\frac{\hat p^2}{2m}+V(\hat x),\qquad \hat p=-\ii\hbar\frac{\partial}{\partial x}$
Time evolution of a stationary statesource
$\Psi(x,t)=\psi(x)\,\ee^{-\ii Et/\hbar}$
Only an overall phase is added; |Ψ|² does not change with time

Probability and expectation values

Born rulesource
$P(a\le x\le b)=\int_a^b|\Psi(x,t)|^2\,\dd x$
Normalisation conditionsource
$\int_{-\infty}^{\infty}|\Psi(x,t)|^2\,\dd x=1$
Probability current densitysource
$j=\frac{\hbar}{2m\ii}\left(\Psi^*\frac{\partial\Psi}{\partial x}-\Psi\frac{\partial\Psi^*}{\partial x}\right)$
Satisfies the continuity equation ∂ρ/∂t + ∂j/∂x = 0
Expectation valuesource
$\langle \hat A\rangle=\int \Psi^*\hat A\,\Psi\,\dd x$
Uncertainty principlesource
$\sigma_x\sigma_p\ge\frac{\hbar}{2}$
Ehrenfest's theoremsource
$\frac{\dd\langle x\rangle}{\dd t}=\frac{\langle p\rangle}{m},\qquad \frac{\dd\langle p\rangle}{\dd t}=\left\langle-\frac{\partial V}{\partial x}\right\rangle$
Expectation values obey Newton's equations

Exact solutions in one dimension

Infinite square wellsource
$\psi_n=\sqrt{\tfrac{2}{L}}\sin\frac{n\pi x}{L},\qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}$
Harmonic-oscillator levelssource
$E_n=\left(n+\tfrac12\right)\hbar\omega,\qquad n=0,1,2,\dots$
Harmonic-oscillator ground statesource
$\psi_0=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\ee^{-m\omega x^2/2\hbar}$
Ladder operatorssource
$\hat a_\pm=\frac{1}{\sqrt{2\hbar m\omega}}(\mp \ii\hat p+m\omega\hat x),\qquad [\hat a_-,\hat a_+]=1$
Gaussian wave packetsource
$\Psi(x,0)=(2\pi\sigma^2)^{-1/4}\ee^{-(x-x_0)^2/4\sigma^2}\ee^{\ii k_0x}$
Free-particle dispersion relationsource
$\omega(k)=\frac{\hbar k^2}{2m},\qquad v_g=\frac{\dd\omega}{\dd k}=\frac{\hbar k}{m}$
The group velocity is twice the phase velocity
Rectangular-barrier transmission (E < V₀)source
$T=\left[1+\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1},\quad \kappa=\frac{\sqrt{2m(V_0-E)}}{\hbar}$
Thick-barrier approximationsource
$T\approx\frac{16E(V_0-E)}{V_0^2}\ee^{-2\kappa a}$
For κa ≫ 1; the exponential factor dominates

Constants and conversions

Reduced Planck constant
$\hbar=1.054\,571\times10^{-34}\ \text{J·s}=6.582\times10^{-16}\ \text{eV·s}$
Electron mass
$m_e=9.109\times10^{-31}\ \text{kg}=511\ \text{keV}/c^2$
Bohr radius
$a_0=\frac{4\pi\varepsilon_0\hbar^2}{m_ee^2}=0.529\ \text{Å}$
Hydrogen ground-state energy
$E_1=-13.606\ \text{eV}$
Fine-structure constant
$\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c}\approx\frac{1}{137.036}$
Handy conversions
$\hbar c=197.3\ \text{eV·nm},\qquad k_BT\big|_{300\text{K}}=25.9\ \text{meV}$