This repository contains a Python-based simulation of Schrödinger’s cat, a famous quantum mechanics thought experiment, using the QuTiP library.

The simulation visualizes the quantum superposition of a cat being alive and dead in phase space via the Wigner function, showcasing:

• Coherent evolution
• Decoherence
• Wave function collapse

Erwin Schrödinger’s 1935 thought experiment illustrates the paradoxical nature of quantum superposition when applied to macroscopic objects:

• A cat is sealed in a box with a radioactive atom, a Geiger counter, and a vial of poison.
• If the atom decays (50% probability), the poison is released → the cat dies.
• Until observed, quantum mechanics suggests the cat exists in a superposition of alive and dead.

This simulation models a quantum analog of the cat state using a coherent state superposition in a harmonic oscillator, visualized through the Wigner function in phase space.

It demonstrates:

• Coherent evolution under a Kerr Hamiltonian → twisting interference patterns
• Decoherence due to environmental interactions → fading quantum interference
• Interactive collapse via key press → mimicking measurement

The simulation is based on a quantum harmonic oscillator with Hilbert space dimension:

$$ N = 30 $$

The Schrödinger cat state is a superposition of two coherent states:

$$ |\psi_\text{cat}\rangle = \frac{1}{\sqrt{2 ,(1 + e^{-2|\alpha|^2})}} \Big( |\alpha\rangle + |-\alpha\rangle \Big), \quad \alpha = 2.0 $$

where $|\alpha\rangle$ and $|-\alpha\rangle$ are coherent states with amplitudes $\alpha$ and $-\alpha$.

The corresponding density matrix is:

$$ \rho_0 = |\psi_\text{cat}\rangle \langle \psi_\text{cat}| $$

The Wigner function $W(x,p)$ represents the quantum state in phase space, computed over a grid:

$$ x, p \in [-5, 5] $$

• Two Gaussian peaks → “Alive” ($x \approx 2$) and “Dead” ($x \approx -2$)
• Interference fringes → signature of quantum superposition

Decomposition of the Wigner function:

$$ W(x,p) = \frac{1}{\mathcal{N}^2} \Big[ W_\alpha(x,p) + W_{-\alpha}(x,p) + W_\text{interf}(x,p) \Big] $$

where:

• $W_\alpha(x,p)$ = Wigner function of $|\alpha\rangle$
• $W_{-\alpha}(x,p)$ = Wigner function of $|-\alpha\rangle$
• $W_\text{interf}(x,p)$ = interference term
• $\mathcal{N}^2 = 2(1 + e^{-2|\alpha|^2})$

The state evolves under a Kerr Hamiltonian:

$$ H_\text{Kerr} = \kappa , (a^\dagger a)^2, \quad \kappa = 0.1 $$

• Non-linear shearing causes interference fringes to twist into spirals
• Gaussian blobs distort in phase space

After resetting to $\rho_0$, decoherence is applied via amplitude damping:

$$ c = \sqrt{\gamma} , a, \quad \gamma = 0.05 $$

• No Hamiltonian applied ($H = 0$)
• Interference fringes fade away, leaving stationary blobs
• System resembles a classical mixture:

$$ \rho_\text{decoh} \approx \frac{1}{2} \Big( |\alpha\rangle \langle \alpha| + |-\alpha\rangle \langle -\alpha| \Big) $$

Pressing “o” collapses the wave function to:

$$ |\psi_\text{cat}\rangle \to \begin{cases} |\alpha\rangle & \text{"Alive"} \\ |-\alpha\rangle & \text{"Dead"} \end{cases} $$

The plot updates with a single labeled blob: Alive or Dead.

• t = 0 → 2 (Static Display): Two blobs (“Alive” at $x \approx 2$, “Dead” at $x \approx -2$) with straight interference fringes.

• t = 2 → 10 (Coherent Evolution): Kerr Hamiltonian twists fringes into spirals, blobs distort.

• t = 10 → 20 (Decoherence): Fringes fade, blobs remain stationary → classical mixture.

• Collapse (press “o”): Single blob remains, labeled Alive or Dead.

• Copenhagen: Collapse occurs on measurement (“o” key)
• Decoherence: Environmental interaction destroys interference (phase 2)
• Other views: Many-Worlds, Bohmian Mechanics, QBism also consistent but not explicitly modeled

The following timeline summarizes key milestones in quantum mechanics, highlighting contributions directly relevant to Schrödinger’s Cat (🐱) and tools used in modern simulations like the one described above (🛠️).

Legend:

• 🐱: Directly relevant to Schrödinger’s Cat (superposition, measurement, entanglement).
• 🛠️: Relevant as a tool for modern simulations (e.g., wave mechanics, Wigner function, or decoherence used in QuTiP visualizations).

• N. Lambert et al., QuTiP 5: The Quantum Toolbox in Python, arXiv:2412.04705 (December 6, 2024). https://arxiv.org/abs/2412.04705

• QuTiP: https://qutip.org

• Schrödinger, E. (1980). The present situation in quantum mechanics. (J. D. Trimmer, Trans.).
  Proceedings of the American Philosophical Society, 124(5), 323–338. (Original work published 1935)

• Wigner, E. P. (1932). On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40, 749–759.

• Zeh, H. D. (1970). On the Interpretation of Measurement in Quantum Theory. Foundations of Physics 1, 69–76.

• Zurek, W. H. (2003). Decoherence and the Transition from Quantum to Classical. arXiv:quant-ph/0306072v1. https://doi.org/10.48550/arXiv.quant-ph/0306072

• Becker R. (2025). Seeing Quantum Weirdness. Medium.
  https://medium.com/@ratwolf/seeing-quantum-weirdness-e977d97a3214

A: This is a visualization effect to highlight the transition, not a physical change. The darker red results from the color scaling (vmin and vmax based on the maximum absolute Wigner value), which emphasizes the remaining amplitude after interference fades.

A: In the simulation, amplitude damping with gamma = 0.05 may cause a slight contraction or shift of the initial coherent states toward the origin due to energy loss, a physical decoherence effect. In reality, this effect might be much smaller, depending on the physical system's decoherence rate.

A: Yes, the sparse fringes reflect the Wigner function’s quantum interference for alpha = 2.0. Adjust alpha or grid size (x, p) in the parameters to explore.

A: With 200 timesteps over 20 units, dt (~0.1) is sufficient. Increase timesteps for higher precision

A: In this simulation, amplitude damping is the default decoherence model, implemented with a collapse operator (c_ops_decoherence) and a damping rate of gamma = 0.05. However, you can modify c_ops_decoherence in the code to include other decoherence models, such as dephasing, to explore different dynamics.

A: Yes, QuTiP’s wigner ensures normalization. The vmin and vmax in contourf capture the full range—see the plotting section.

A: Pressing 'o' randomly selects a pure state (psi1 or psi2), simulating measurement per the Copenhagen interpretation.

A: Yes, increase N (currently 30) or x, p grid (currently 130) in the parameters, though it may slow performance.