1D Quantum Harmonic Oscillator with Neural Quantum States

In this example, we solve the 1D harmonic oscillator, $$i\partial_t\Psi = \left[ \frac{1}{2}\partial^2_x + V(x)\right]\Psi$$
We employ a NQS with complex parameters.
An overview can be found in the Poster. A preprint is available in PDF and in arXiv.

The Neural Quantum State (NQS)

The default NQS is a neural network (NN) consisting of 1 input and 1 output neurons and a single hidden layer with 5 neurons. Accounting for the biases, there is a total of 16 complex parameters, $\vec{\theta} = {\theta_0,\theta_1,\dots,\theta_{15}}$.
In all neurons, the activation function is a Sigmoid.
The output of the NN is exponenciated and then interpreted as $\ln \Psi_{\vec{\theta}}(\vec{x})$.

The physical system

The physical system is confined in a box of size $L=16$.
The spatial grid $\vec{x}$ is discretized in $n=100$ points.
The harmonic trapping potential $V(x)=\frac{1}{2}\omega^2(x-x_0)^2$ with $\Omega=1$.
(note: we employ harmonic oscillator units)

The coherent state

Initially, the ground state is found with the trapping potential set at $x_0=1$. Then, the dynamics are initialized when instantanoulsy displacing the trapping potential to $x_0=0$.

The dynamics of the coherent state can be reproduced running main_oscilaltions.py

The breathing mode

Initially, the ground state is found with the trapping potential set at $x_0=0$. Then, the dynamics are initialized when instantanoulsy changing the trapping frequency to $\omega=0.5$.

The dynamics of the breathing mode can be reproduced running main_breathing.py