5. Decoherence

Decoherence explains the emergence of classical behavior from quantum systems through interaction with the environment. It resolves the measurement problem by showing how superpositions are destroyed in practice, without invoking wave function collapse.

The Measurement Problem

Quantum mechanics predicts superpositions, yet we observe definite outcomes:

$$|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \quad \xrightarrow{\text{measure}} \quad \begin{cases} |0\rangle & \text{prob } 1/2 \\ |1\rangle & \text{prob } 1/2 \end{cases}$$

Questions:

Why don't we see macroscopic superpositions (Schrödinger's cat)?
What constitutes a "measurement"?
When does the wave function "collapse"?

Decoherence answer: Interaction with environment rapidly destroys quantum coherence for macroscopic systems, making superpositions effectively classical mixtures

System-Environment Interaction

Total Hamiltonian:

$$\hat{H} = \hat{H}_S + \hat{H}_E + \hat{H}_{int}$$

System + Environment + Interaction

Initial state (product state):

$$|\Psi(0)\rangle = |\psi_S\rangle \otimes |E_0\rangle$$

After interaction, system becomes entangled with environment:

$$|\Psi(t)\rangle = \sum_i c_i |i\rangle_S \otimes |E_i(t)\rangle$$

Different system states $|i\rangle_S$ become correlated with different environment states $|E_i\rangle$

Reduced Density Matrix Evolution

System's reduced density matrix:

$$\hat{\rho}_S(t) = \text{Tr}_E[|\Psi(t)\rangle\langle\Psi(t)|] = \sum_{i,j} c_i c_j^* \langle E_j(t)|E_i(t)\rangle \, |i\rangle\langle j|$$

Off-diagonal elements (coherences) decay as environment states become orthogonal:

$$\rho_{ij}(t) = c_i c_j^* \langle E_j(t)|E_i(t)\rangle \approx c_i c_j^* e^{-\Gamma_{ij} t}$$

Decoherence rate $\Gamma_{ij}$ depends on system-environment coupling

Result: Pure superposition → Mixed state (classical probability distribution)

$$\hat{\rho}_S(t \to \infty) = \sum_i |c_i|^2 |i\rangle\langle i|$$

Pointer States

Not all bases decohere equally - certain "pointer states" are robust:

Pointer basis selection: States that minimize entanglement production with environment

$$\hat{H}_{int} = \hat{S} \otimes \hat{E} \quad \Rightarrow \quad \text{eigenstates of } \hat{S} \text{ are pointer states}$$

Examples:

Position: For scattering interactions ($\hat{H}_{int} \sim \hat{x} \otimes \hat{E}$), position eigenstates are robust
Energy: For thermal baths, energy eigenstates decohere slowly
Spin: For magnetic environments, spin basis is preferred

Key insight: Classical observables emerge as those whose eigenstates are pointer states!

Decoherence Timescales

Typical decoherence time for position superposition:

$$\tau_D \sim \frac{\hbar}{\lambda^2 n_{\text{env}} \sigma v}$$

Where $\lambda$ is separation, $n_{\text{env}}$ is environment density, $\sigma$ is scattering cross section, $v$ is velocity

Examples:

Dust grain (1 μm) in air: $\tau_D \sim 10^{-31}$ s
Large molecule in vacuum: $\tau_D \sim 10^{-3}$ s
Electron in cavity: $\tau_D \sim 1$ s
Photon polarization: $\tau_D \sim$ hours (very robust)

Compare to dynamical timescale $\tau_S = \hbar/\Delta E$:

$$\frac{\tau_D}{\tau_S} \sim 10^{-40} \quad \text{(macroscopic objects)}$$

Decoherence is extraordinarily fast compared to quantum evolution!

Master Equation Approach

For weak system-environment coupling, Lindblad master equation:

$$\frac{d\hat{\rho}_S}{dt} = -\frac{i}{\hbar}[\hat{H}_S, \hat{\rho}_S] + \sum_k \gamma_k \left(\hat{L}_k\hat{\rho}_S\hat{L}_k^\dagger - \frac{1}{2}\{\hat{L}_k^\dagger\hat{L}_k, \hat{\rho}_S\}\right)$$

Lindblad operators $\hat{L}_k$ describe different decoherence channels

Example: Pure dephasing (loss of phase coherence)

$$\hat{L} = \sqrt{\gamma}\hat{\sigma}_z \quad \Rightarrow \quad \rho_{01}(t) = \rho_{01}(0) e^{-\gamma t}$$

Off-diagonal elements decay exponentially

Example: Amplitude damping (energy loss)

$$\hat{L} = \sqrt{\gamma}\hat{\sigma}_- \quad \Rightarrow \quad \text{population decays from } |1\rangle \text{ to } |0\rangle$$

Quantum-to-Classical Transition

Decoherence provides mechanism for classical emergence:

Superposition selection: Environment selects pointer basis (typically position/momentum for macroscopic objects)
Rapid decoherence: Coherences vanish on timescale $\tau_D \ll$ observation time
Effective collapse: State becomes diagonal in pointer basis (classical probability distribution)
Classical dynamics: Surviving density matrix evolves according to classical equations

Important: Decoherence does NOT solve measurement problem completely - it shows why we don't see macroscopic superpositions, but doesn't explain individual measurement outcomes (still need interpretation)

Experimental Demonstrations

1. Cavity QED experiments (Haroche, 2012 Nobel Prize):

Observed gradual decoherence of photon number states in cavity as environment coupling increased

2. Matter-wave interferometry:

Large molecules (C$_{70}$) showed interference fringes that vanished when gas pressure increased (controllable decoherence)

3. Superconducting qubits:

Direct observation of dephasing and relaxation processes, measurement of $T_1$ (amplitude damping) and $T_2$ (dephasing) times

4. Trapped ions:

Creation of Schrödinger cat states and controlled decoherence into classical mixtures

Quantum Error Correction

Fighting decoherence is essential for quantum computing:

Basic idea: Encode logical qubit in multiple physical qubits

\begin{align*} |0_L\rangle &= |000\rangle \\ |1_L\rangle &= |111\rangle \end{align*}

Simple repetition code (can correct bit flips)

Stabilizer codes:

Shor code (9 qubits): corrects arbitrary single-qubit errors
Surface codes: scalable, high threshold (~1% error rate)
Topological codes: protect via global properties

Threshold theorem: If decoherence rate below threshold, arbitrarily long quantum computation possible with polynomial overhead

Decoherence-Free Subspaces

Special subspaces immune to certain decoherence processes:

For collective dephasing $\hat{H}_{int} = (\hat{\sigma}_z^{(1)} + \hat{\sigma}_z^{(2)}) \otimes \hat{E}$:

$$|\psi_{\text{DFS}}\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle)$$

Singlet state is immune (equal and opposite phases cancel)

Applications:

Robust storage of quantum information
Passive error suppression (no active correction needed)
Combine with error correction for enhanced protection

Physical Insight: Decoherence reconciles quantum mechanics with classical experience. It's not a new mechanism but a consequence of entanglement with the environment. The same quantum mechanics that predicts superpositions also predicts their destruction in open systems. This insight revolutionized our understanding of the quantum-classical boundary and is central to quantum technology.

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