Case Study 2: Decoherence — Why Schrodinger's Cat Is Always Dead or Alive

Overview

In 1935, Erwin Schrodinger devised his famous thought experiment to illustrate what he considered an absurd consequence of quantum mechanics: a cat in a superposition of alive and dead. For decades, the puzzle seemed merely philosophical — an uncomfortable question that physicists could safely ignore while computing cross sections and energy levels. But starting in the 1970s and 1980s, a quantitative answer emerged: decoherence, the process by which environmental entanglement destroys macroscopic superpositions on timescales so short that they are unobservable in practice.

This case study traces the Schrodinger cat problem from thought experiment to quantitative physics. We follow the density matrix through the decoherence process step by step, compute decoherence timescales for realistic systems, and confront what decoherence does and does not explain.

Part 1: The Thought Experiment

Schrodinger's Setup (1935)

Schrodinger's original scenario is as follows. A cat is enclosed in a box along with:

1. A radioactive atom with a 50% chance of decaying within one hour
2. A Geiger counter that detects the decay
3. A mechanism that releases poison gas if the Geiger counter clicks

After one hour, the radioactive atom is in a superposition of "decayed" and "not decayed." Because the mechanism couples the atom's state to the cat's fate, quantum mechanics (taken literally) predicts that the cat is in a superposition:

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

Schrodinger's point was not that he believed in zombie cats. His point was that the linearity of quantum mechanics, combined with the entangling interaction between microscopic and macroscopic systems, inevitably leads to macroscopic superpositions — and yet we never observe them.

🔵 Historical Note: Schrodinger wrote to Einstein in 1935: "One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat)..." The paper in which this appeared, "Die gegenwartige Situation in der Quantenmechanik" (Naturwissenschaften, 1935), also introduced the term entanglement (Verschrankung) — making it one of the most consequential papers in the history of physics.

The Problem in Density Matrix Language

The composite state of atom + cat is:

$$\hat{\rho}_{\text{total}} = |\Psi\rangle\langle\Psi| = \frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{pmatrix}$$

in the basis $\{|\text{intact, alive}\rangle, |\text{intact, dead}\rangle, |\text{decayed, alive}\rangle, |\text{decayed, dead}\rangle\}$.

The off-diagonal elements $\hat{\rho}_{14} = \hat{\rho}_{41}^* = 1/2$ represent quantum coherence between "alive" and "dead." If we could observe interference between these branches — for example, in a "cat interferometer" — it would confirm the superposition. The question is: can we?

Part 2: Enter the Environment

The Key Realization

The crucial insight, developed by H. Dieter Zeh (1970), Wojciech Zurek (1981–1982), and others, is that Schrodinger's thought experiment ignores the environment. The cat is not an isolated quantum system. It is immersed in air (approximately $10^{25}$ molecules), bathed in thermal radiation (approximately $10^{13}$ photons per cubic centimeter at room temperature), and gravitationally coupled to everything around it. Each of these environmental degrees of freedom can interact with — and become entangled with — the cat's macroscopic state.

Modeling Decoherence: The Spin-Boson Approach

Let us model the problem carefully. Replace the cat with a two-state system $S$ (alive/dead ↔ $|0\rangle$/$|1\rangle$) and the environment with $N$ two-state systems ("spins") representing air molecules, photons, etc.

The initial state is:

$$|\Psi(0)\rangle = \frac{1}{\sqrt{2}}(|0\rangle_S + |1\rangle_S) \otimes |e_0\rangle_E$$

where $|e_0\rangle_E$ is the initial environment state.

The system-environment interaction causes:

$$|0\rangle_S|e_0\rangle_E \to |0\rangle_S|e_0(t)\rangle_E, \quad |1\rangle_S|e_0\rangle_E \to |1\rangle_S|e_1(t)\rangle_E$$

where $|e_0(t)\rangle$ and $|e_1(t)\rangle$ are the environment states that have been "imprinted" by the system states. By linearity:

$$|\Psi(t)\rangle = \frac{1}{\sqrt{2}}(|0\rangle_S|e_0(t)\rangle_E + |1\rangle_S|e_1(t)\rangle_E)$$

The Reduced Density Matrix

Tracing over the environment:

$$\hat{\rho}_S(t) = \frac{1}{2}\begin{pmatrix} 1 & \langle e_1(t)|e_0(t)\rangle \\ \langle e_0(t)|e_1(t)\rangle & 1 \end{pmatrix}$$

The off-diagonal element — the decoherence factor — is $\Gamma(t) = \langle e_1(t)|e_0(t)\rangle$.

Part 3: Computing the Decoherence Time

The Collisional Decoherence Model

For a massive object in a gas, decoherence occurs through scattering of gas molecules. Each molecule that scatters off the object acquires "which-state" information — its scattered wavefunction depends on the object's position.

Joos and Zeh (1985) derived the decoherence rate for an object of size $a$ in a gas with particle density $n_{\text{gas}}$ and mean thermal velocity $v_{\text{th}}$:

$$\Gamma_{\text{dec}} \approx n_{\text{gas}} \cdot v_{\text{th}} \cdot a^2 \cdot \left(\frac{\Delta x}{\lambda_{\text{th}}}\right)^2$$

where $\Delta x$ is the spatial separation of the two superposed positions and $\lambda_{\text{th}} = \hbar/(m_{\text{gas}}v_{\text{th}})$ is the thermal de Broglie wavelength of the gas particles.

The decoherence time is $\tau_d = 1/\Gamma_{\text{dec}}$.

Numerical Estimates

Let us compute $\tau_d$ for several systems in air at room temperature ($T = 300$ K, $n_{\text{gas}} \approx 2.5 \times 10^{25}$ m$^{-3}$, $m_{\text{gas}} = 4.8 \times 10^{-26}$ kg for N$_2$, $v_{\text{th}} \approx 500$ m/s, $\lambda_{\text{th}} \approx 2 \times 10^{-11}$ m):

Case 1: An electron ($a \sim 10^{-15}$ m) in a superposition of $\Delta x = 10^{-10}$ m (atomic scale):

$$\Gamma_{\text{dec}} \approx 2.5 \times 10^{25} \times 500 \times 10^{-30} \times \left(\frac{10^{-10}}{2\times10^{-11}}\right)^2 \approx 3 \times 10^{-1}\;\text{s}^{-1}$$

$$\tau_d \approx 3\;\text{s}$$

This is slow — quantum coherence of an electron in air can survive for seconds. This is why we can do electron diffraction experiments.

Case 2: A large molecule (C$_{70}$ fullerene, $a \sim 10^{-9}$ m) in a superposition of $\Delta x = 10^{-7}$ m:

$$\Gamma_{\text{dec}} \approx 2.5 \times 10^{25} \times 500 \times 10^{-18} \times \left(\frac{10^{-7}}{2\times10^{-11}}\right)^2 \approx 3 \times 10^{14}\;\text{s}^{-1}$$

$$\tau_d \approx 3 \times 10^{-15}\;\text{s}$$

Decoherence is fast. This is why fullerene interference experiments must be done in high vacuum.

Case 3: A dust grain ($a \sim 10^{-5}$ m) in a superposition of $\Delta x \sim a$:

$$\Gamma_{\text{dec}} \approx 2.5 \times 10^{25} \times 500 \times 10^{-10} \times \left(\frac{10^{-5}}{2\times10^{-11}}\right)^2 \approx 3 \times 10^{30}\;\text{s}^{-1}$$

$$\tau_d \approx 3 \times 10^{-31}\;\text{s}$$

This is absurdly fast — far shorter than any conceivable measurement time.

Case 4: A cat ($a \sim 0.3$ m) in a superposition of alive and dead ($\Delta x \sim 0.3$ m):

$$\Gamma_{\text{dec}} \approx 2.5 \times 10^{25} \times 500 \times 0.09 \times \left(\frac{0.3}{2\times10^{-11}}\right)^2 \approx 10^{51}\;\text{s}^{-1}$$

$$\tau_d \approx 10^{-51}\;\text{s}$$

The cat's superposition decoheres in $10^{-51}$ seconds. The age of the universe is $4 \times 10^{17}$ seconds. Schrodinger's cat is never in a superposition in any operationally meaningful sense — the environment destroys it before a single photon can bounce off the cat.

💡 Key Insight: Decoherence is not a subtle effect. It is not a small correction. For macroscopic objects, it is a $10^{30}$-fold acceleration of the transition from quantum to classical. This is why the macroscopic world looks classical — not because of any deficiency in quantum mechanics, but because the environment is an extraordinarily effective measurement apparatus.

Summary Table

Part 4: Experimental Tests of Decoherence

The Brune et al. Experiment (1996)

One of the most beautiful demonstrations of decoherence was performed by Serge Haroche's group at the Ecole Normale Superieure in Paris. They created a "Schrodinger cat state" of the electromagnetic field in a microwave cavity — a superposition of two coherent states with opposite phases:

$$|\psi_{\text{cat}}\rangle \propto |\alpha\rangle + |-\alpha\rangle$$

By sending probe atoms through the cavity at successive times, they observed the coherences between $|\alpha\rangle$ and $|-\alpha\rangle$ decaying exponentially, with a decoherence rate proportional to $|\alpha|^2$ — exactly as predicted by the theory.

This experiment directly verified: 1. Schrodinger cat states can be created (in carefully controlled microscopic systems) 2. They decohere at a rate that increases with the "size" of the superposition 3. The decoherence time matches theoretical predictions quantitatively

Haroche shared the 2012 Nobel Prize in Physics for this work.

Matter-Wave Interferometry

The group of Markus Arndt in Vienna has pushed matter-wave interference to increasingly large molecules — from C$_{60}$ (1999) to molecules with over 2000 atoms and masses above 25,000 daltons (2019). Each experiment requires progressively better vacuum and vibration isolation to prevent decoherence.

The experimental progression beautifully confirms the decoherence model: - Interference visibility decreases with increasing gas pressure (more collisional decoherence) - Interference visibility decreases with increasing temperature (more thermal photon scattering) - Interference visibility decreases with increasing molecular size (larger cross section for environmental scattering)

Superconducting Qubits

Modern superconducting qubits (transmons) operate at $T \approx 10$ mK in dilution refrigerators, achieving coherence times of $T_2 \approx 100$--$300\;\mu$s (as of 2024). The remaining decoherence comes from:

• Residual thermal photons in the microwave cavity
• Two-level system (TLS) defects in the Josephson junction and substrate
• Quasiparticle tunneling across the junction
• Coupling to the measurement apparatus

Each decoherence mechanism leaves a distinct signature in the time dependence of the density matrix, allowing experimentalists to diagnose and mitigate individual noise sources.

Part 5: What Decoherence Does Not Explain

The Measurement Problem Persists

After decoherence, the density matrix of the cat is:

$$\hat{\rho}_{\text{cat}} \approx \frac{1}{2}|\text{alive}\rangle\langle\text{alive}| + \frac{1}{2}|\text{dead}\rangle\langle\text{dead}|$$

This looks like a classical mixture — a coin that has already been flipped. But the total state of cat + environment is still a pure, entangled state:

$$|\Psi\rangle \approx \frac{1}{\sqrt{2}}(|\text{alive}\rangle|e_{\text{alive}}\rangle + |\text{dead}\rangle|e_{\text{dead}}\rangle)$$

Decoherence has converted a local superposition into a global entanglement. It has not eliminated any branch — both "alive" and "dead" persist in the total state. The question of why we observe one outcome rather than both remains.

Different interpretations address this differently:

Copenhagen interpretation: Collapse occurs (somehow) and selects one outcome. Decoherence explains why the alternatives don't interfere, but the collapse postulate is still needed.

Many-worlds interpretation: Both outcomes occur in different "branches" of the universal wavefunction. Decoherence explains why the branches don't interfere — they are orthogonal in the environmental Hilbert space — but both branches are equally real.

Bohmian mechanics: The particle always had a definite position. Decoherence explains why the empty wave packet (the unoccupied branch) becomes dynamically irrelevant — it can no longer guide the particle back.

QBism: The density matrix represents the agent's personal beliefs about future experiences. Decoherence updates those beliefs in a particular way, but there is no objective "collapse."

⚖️ Interpretation: Decoherence is arguably the most important physics result in quantum foundations in the past 50 years. But it is interpretation-neutral: every major interpretation accepts it, and none is eliminated by it. The measurement problem remains open. If someone tells you decoherence "solves" the measurement problem, they are confusing necessary with sufficient. We discuss this at length in Chapter 28.

Discussion Questions

1. Schrodinger's original thought experiment was meant as a reductio ad absurdum of the quantum formalism. Given what we now know about decoherence, does the thought experiment still achieve its rhetorical purpose? Or does decoherence defuse it?

2. The decoherence rate for a dust grain in air is $\sim 10^{30}$ s$^{-1}$, but in a perfect vacuum at $T = 0$ it would be zero. In principle, could we maintain a dust grain in a quantum superposition in perfect isolation? What practical challenges would arise?

3. Some physicists argue that decoherence makes the many-worlds interpretation more plausible (because it explains why branches don't interact), while others argue it makes it less necessary (because we don't need parallel worlds to explain why we don't see superpositions). Evaluate both arguments.

4. The 2019 Arndt group experiment demonstrated quantum interference with molecules of mass $\sim 25{,}000$ daltons. What is the largest object for which quantum interference might plausibly be observed? What technological barriers must be overcome? (Research the MAQRO satellite proposal.)

5. In the Haroche experiment, the decoherence of the "cat state" was observed by sending probe atoms into the cavity one at a time. Each probe partially collapses the cat state. How does this progressive measurement relate to the continuous decoherence described in this case study?

Further Exploration

• Quantitative exercise: Repeat the decoherence timescale calculation for a superposition of two positions of the Moon (mass $\sim 7 \times 10^{22}$ kg, radius $\sim 2 \times 10^6$ m) separated by 1 meter, with the cosmic microwave background ($T = 2.7$ K) as the sole environment. Show that even in the near-vacuum of space, decoherence is absurdly fast.

• Paper study: Read Zurek's landmark paper "Decoherence, einselection, and the quantum origins of the classical" (Reviews of Modern Physics, 2003). Summarize the key arguments in 2-3 pages, focusing on the concept of einselection (environment-induced superselection).

• Computational project: Simulate a qubit interacting with a bath of $N = 10$ environmental qubits. Start the system in a superposition and the bath in a product state. Evolve the total system unitarily and compute the reduced density matrix of the system qubit at each time step. Plot the decoherence of the off-diagonal elements and verify that it becomes faster as $N$ increases. Use the code framework from code/project-checkpoint.py.