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https://github.com/ratwolfzero/schroedingers_cat

Schrödinger’s Cat Simulation
https://github.com/ratwolfzero/schroedingers_cat

coherence decoherence probability-density python quantum-physics qutip schrodinger-equation schroedingers-cat wave-function wave-function-collapse

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Schrödinger’s Cat Simulation

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# Schrödinger’s Cat Simulation



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**



![Simulation](schrodinger_cat.gif)



---



## 📖 Overview



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



---



## 🧮 Mathematical Background



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



$$

N = 30

$$



### Initial Cat State



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}|

$$



---



### Wigner Function



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

* $\mathcal{N}^2 = 2(1 + e^{-2|\alpha|^2})$



![initial](initial_state.png)



---



## Simulation Phases



### 1. Coherent Evolution (t = 2 → 10)



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



![Coherent](coherent_evolution.png)



---



### 2. Decoherence (t = 10 → 20)



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)

$$



![Decoherence](decoherence.png)



---



### 3. Collapse (Interactive Measurement)



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**.



![Collapse](alive.png)



---



## 🔍 Interpreting the Results



* **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.



---



## 🌌 Quantum Interpretations



* **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



---



## Quantum Mechanics Timeline



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 (🛠️).



| Relevance       | Year | Figure                  | Contribution                                                      |

| --------------- | ---- | ----------------------- | ----------------------------------------------------------------- |

|                 | 1900 | Planck                  | Quantum Hypothesis ("Revolutionary against his will")             |

|                 | 1905 | Einstein                | Photoelectric Effect (light as quanta)                            |

|                 | 1913 | Bohr                    | Atomic Model (quantized orbits)                                   |

|                 | 1925 | Heisenberg              | Matrix Mechanics (observables, not orbits)                        |

| 🐱🛠️           | 1926 | Schrödinger             | Wave Mechanics (wavefunction dynamics)                            |

| 🐱              | 1926 | Born                    | Probabilistic Interpretation (wavefunction → probability)         |

| 🐱              | 1927 | Bohr/Heisenberg         | Copenhagen Interpretation (measurement & observer)                |

|                 | 1928 | Dirac                   | Uniting QM with special relativity (prediction of antimatter)     |

| 🐱              | 1932 | von Neumann             | Mathematical Foundations (axioms, measurement theory)             |

| 🐱🛠️           | 1932 | Wigner                  | Phase-space interpretation (Wigner function, quasi-probabilities) |

| 🐱              | 1935 | Einstein-Podolsky-Rosen | EPR Paradox (QM works, but is it complete? — still debated)       |

| 🐱              | 1935 | Schrödinger             | Schrödinger’s Cat (paradox of superposition)                      |

| 🐱🛠️           | 1970 | Zeh                     | Decoherence Theory (quantum-classical transition)                 |

| 🐱🛠️           | 1980s–2003 | Zurek           | Decoherence and Quantum-Classical Transition (pointer states)     |



**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).



---



## 📚 References



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

* QuTiP: [https://qutip.org](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.* [https://doi.org/10.48550/arXiv.quant-ph/0306072](https://doi.org/10.48550/arXiv.quant-ph/0306072)

* Becker R. (2025). Seeing Quantum Weirdness. Medium.  

  



---



## Schrödinger's Cat Simulation technical FAQ



### Q: Why are the red blobs darker during decoherence?



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.



### Q: Why do the blobs shift slightly during decoherence?



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.



### Q: Are the interference fringes correct?



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.



### Q: Is the time step (dt) accurate?



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



### Q: Is amplitude damping the only decoherence model?



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.



### Q: Is the Wigner function properly normalized?



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



### Q: How does collapse work?



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



### Q: Can I improve accuracy with higher N or grid resolution?



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