Why We Don't See Quantum Effects" chapter: 33 type: case-study case_study_number: 1

Case Study 1: Decoherence --- Why We Don't See Quantum Effects

The Central Puzzle

In 1935, Erwin Schrodinger proposed his famous thought experiment: a cat sealed in a box with a radioactive atom, a Geiger counter, and a vial of poison. If the atom decays, the counter triggers, the vial breaks, and the cat dies. Quantum mechanics seems to predict that before we open the box, the cat is in a superposition of alive and dead. Yet we never experience such macroscopic superpositions. Why not?

For decades, this was treated primarily as a philosophical problem --- a puzzle about interpretation rather than physics. The development of decoherence theory, beginning with H. Dieter Zeh in the 1970s and Wojciech Zurek in the 1980s, transformed it into a quantitative physical question with a precise answer.

The Quantitative Argument

Setting Up the Problem

Consider a simplified "Schrodinger's cat" scenario: a single two-level atom coupled to a macroscopic detector (the cat, effectively). The atom starts in a superposition:

$$|\psi_{\text{atom}}\rangle = \frac{1}{\sqrt{2}}(|\text{undecayed}\rangle + |\text{decayed}\rangle)$$

If the detector amplifies faithfully, the combined state becomes:

$$|\Psi\rangle = \frac{1}{\sqrt{2}}(|\text{undecayed}\rangle|\text{alive}\rangle + |\text{decayed}\rangle|\text{dead}\rangle)$$

This is a macroscopic superposition --- a "cat state." In a closed, perfectly isolated system, this state would persist indefinitely, and interference between the two branches would be observable in principle.

The Environment Enters

But the cat is not isolated. It consists of roughly $10^{26}$ atoms, each interacting with thermal photons, air molecules, and phonons in the box walls. Within an extraordinarily short time, each of these environmental particles becomes correlated with the macroscopic state:

$$|\Psi\rangle \to \frac{1}{\sqrt{2}}(|\text{undecayed}\rangle|\text{alive}\rangle|\mathcal{E}_{\text{alive}}\rangle + |\text{decayed}\rangle|\text{dead}\rangle|\mathcal{E}_{\text{dead}}\rangle)$$

where $|\mathcal{E}_{\text{alive}}\rangle$ and $|\mathcal{E}_{\text{dead}}\rangle$ represent the states of the billions of environmental particles.

The Overlap Goes to Zero

The critical quantity is the overlap between the two environmental states:

$$\langle\mathcal{E}_{\text{dead}}|\mathcal{E}_{\text{alive}}\rangle$$

Each individual environmental particle (a photon, an air molecule) acquires only a tiny correlation with the cat's state. If the $k$-th environmental particle has states $|e_k^{(\text{alive})}\rangle$ and $|e_k^{(\text{dead})}\rangle$ with overlap $|\langle e_k^{(\text{dead})}|e_k^{(\text{alive})}\rangle| = 1 - \epsilon_k$ where $\epsilon_k \ll 1$, then for $N$ independent environmental particles:

$$|\langle\mathcal{E}_{\text{dead}}|\mathcal{E}_{\text{alive}}\rangle| = \prod_{k=1}^N (1 - \epsilon_k) \approx e^{-\sum_k \epsilon_k}$$

Even if each $\epsilon_k$ is tiny (say $10^{-20}$), the sum over $N \sim 10^{26}$ particles gives $\sum_k \epsilon_k \sim 10^{6}$, making the overlap $e^{-10^6} \approx 0$. This is decoherence: the off-diagonal elements of the reduced density matrix vanish superexponentially fast.

Timescale Estimates

Joos and Zeh (1985) computed the decoherence rate for a sphere of radius $a$ at temperature $T$ due to scattering of thermal photons:

$$\Lambda_{\text{photons}} \approx 10^{36} \left(\frac{a}{1\,\text{cm}}\right)^6 \left(\frac{T}{300\,\text{K}}\right)^9 \,\text{m}^{-2}\text{s}^{-1}$$

For a cat-sized object ($a \sim 10$ cm) at room temperature with a superposition separation $\Delta x \sim 10$ cm:

$$\tau_{\text{dec}} = \frac{1}{\Lambda_{\text{photons}} \cdot (\Delta x)^2} \sim \frac{1}{10^{42} \times 10^{-2}} = 10^{-40}\,\text{s}$$

This is approximately $10^{-40}$ seconds --- a time so short that it is essentially zero for any conceivable experiment. The superposition is destroyed before the first photon has even completed a single oscillation cycle.

The Experimental Frontier

While macroscopic superpositions are destroyed instantaneously, researchers have progressively pushed the boundary by working with mesoscopic systems in carefully controlled environments.

Cavity QED: Watching a Cat Die

The landmark experiment by Brune, Hagley, Dreyer, Maitre, Maali, Wunderlich, Raimond, and Haroche at the Ecole Normale Superieure in Paris (1996) remains one of the most elegant demonstrations of decoherence. They created a "Schrodinger kitten" --- a superposition of two coherent states of a microwave field trapped in a superconducting cavity:

$$|\psi_{\text{cat}}\rangle = \frac{1}{\sqrt{2}}(|\alpha\rangle + |-\alpha\rangle)$$

By sending probe atoms through the cavity at successive times, they measured the decay of interference fringes. The key result: the decoherence rate was proportional to $|\alpha|^2$ (the square of the "cat size"), exactly as predicted by theory. Larger cats decohered faster, with the rate scaling precisely as $\Gamma_{\text{dec}} = 2\kappa|\alpha|^2$ where $\kappa$ is the single-photon loss rate.

Molecule Interferometry: Quantum Behavior at the Boundary

The Vienna group led by Markus Arndt and Anton Zeilinger has systematically tested quantum interference with increasingly large molecules:

The 2003 C$_{70}$ experiment by Hornberger et al. was particularly illuminating: by varying the background gas pressure, they observed the interference fringes degrade quantitatively as predicted by decoherence theory. This was the first controlled demonstration of the transition from quantum to classical behavior as environmental coupling increased.

Mechanical Oscillators: Quantum Drums and Mirrors

In 2010, O'Connell et al. cooled a micrometer-scale mechanical oscillator (visible to the naked eye) to its quantum ground state and placed it in a quantum superposition of zero and one phonon. More recent experiments with optomechanical systems have achieved superpositions involving billions of atoms, approaching the regime where gravitational decoherence --- decoherence caused by the gravitational field itself --- might be detectable.

Einselection in Action

Why Position is Special

Why does the environment "choose" to measure position rather than, say, momentum or energy?

The answer lies in the form of the interaction. For macroscopic objects, the dominant environmental interactions (electromagnetic scattering, gravitational forces) depend on the positions of particles. The system-environment interaction Hamiltonian has the approximate form:

$$\hat{H}_{\text{int}} \sim \sum_k g_k\,\hat{x}\otimes\hat{B}_k^{(E)}$$

where $\hat{x}$ is the position operator of the system. The pointer states --- the states that survive decoherence --- are the eigenstates of $\hat{x}$, or more precisely, narrow Gaussian wavepackets in phase space (since position and momentum cannot both be perfectly defined due to the Heisenberg uncertainty principle).

This is einselection at work: the environment selects position as the "classical" observable. We observe chairs in definite locations, not in definite momentum states, because the electromagnetic interactions that mediate our observations are position-dependent.

The Quantum-to-Classical Boundary

There is no sharp boundary between quantum and classical. The transition is smooth, governed by the ratio of decoherence time to the timescale of observation:

• $\tau_{\text{dec}} \gg \tau_{\text{obs}}$: Quantum regime. Coherent superpositions persist long enough to be observed.
• $\tau_{\text{dec}} \sim \tau_{\text{obs}}$: Mesoscopic regime. Decoherence is observable and controllable. This is where the most interesting physics happens.
• $\tau_{\text{dec}} \ll \tau_{\text{obs}}$: Classical regime. Decoherence is so fast that all observations show classical behavior.

For everyday objects, $\tau_{\text{dec}}/\tau_{\text{obs}} \sim 10^{-40}$. We are deep, deep in the classical regime.

What Decoherence Does Not Explain

It is crucial to understand the limits of the decoherence explanation:

1. The outcome problem: After decoherence, the density matrix is diagonal in the pointer basis: $\hat{\rho} \approx |\alpha|^2|\text{alive}\rangle\langle\text{alive}| + |\beta|^2|\text{dead}\rangle\langle\text{dead}|$. This looks like a classical probability distribution, but it is derived from a pure entangled state of system+environment. Whether the "other branch" still exists in some sense is an interpretive question that decoherence cannot settle.

2. The Born rule: Decoherence explains why the pointer basis is selected, but it does not derive the probability rule $p_k = |\langle k|\psi\rangle|^2$ from first principles. Attempts to do so (Zurek's "envariance" argument, the Deutsch-Wallace decision-theoretic approach) remain debated.

3. The preferred factorization: Decoherence assumes a particular division of the universe into "system" and "environment." Why this division exists, and whether it is fundamental or emergent, is an open question.

Discussion Questions

1. If decoherence fully explains why macroscopic superpositions are unobservable, does the measurement problem still exist? What additional explanatory work, if any, remains?

2. The decoherence timescale for a bowling ball is $\sim 10^{-42}$ s. Is it meaningful to say the bowling ball was "ever in a superposition" if the superposition is destroyed faster than any physical process could detect it?

3. Molecule interferometry has demonstrated quantum behavior for objects with thousands of atoms. Is there a fundamental limit to how large a quantum superposition can be, or is it purely a matter of technology?

4. How does the concept of quantum Darwinism change our understanding of "classical reality"? If classical information is defined by its redundancy in the environment, does this make the classical world more or less "real"?

5. Some physicists (Penrose, Diosi) propose that gravity causes objective collapse of the wavefunction at a certain mass scale. How would this differ experimentally from "standard" environmental decoherence? What experiments could test this hypothesis?

Connections to Other Chapters

• Chapter 23 (Density Operators): The formalism of partial traces and reduced density matrices is the mathematical backbone of decoherence theory.
• Chapter 35 (Quantum Foundations): Deeper exploration of the interpretive questions raised by decoherence --- many-worlds, objective collapse, QBism.
• Chapter 34 (Quantum Information): The practical consequences of decoherence for quantum computation and communication.
• Chapter 7 (Uncertainty Principle): Pointer states as minimum-uncertainty wavepackets connect decoherence to the uncertainty principle.

The classical world is not a separate realm governed by different laws. It is the quantum world, viewed through the lens of environmental entanglement --- a world where the environment has already performed the measurement for us.