Quick Start Guide

Simulation Set up

Start by importing the module:

import trottersuzuki as ts

To set up the simulation of a quantum system, we need only a few lines of code. First of all, we create the lattice over which the physical system is defined. All information about the discretized space is collected in a single object. Say we want a squared lattice of 300x300 nodes, with a physical area of 20x20, then we have to specify these in the constructor of the Lattice class:

grid = ts.Lattice2D(300, 20.)

The object grid defines the geometry of the system and it will be used throughout the simulations. Note that the origin of the lattice is at its centre and the lattice points are scaled to the physical locations. The actual transformation is exposed to the user via the function center_coordinates.

The physics of the problem is described by the Hamiltonian. A single object is going to store all the information regarding the Hamiltonian. The module is able to deal with two physical models: Gross-Pitaevskii equation of a single or two-component wave function, namely (in units \(\hbar=1\)):

\[i \frac{\partial}{\partial t} \psi(t) = H \psi(t)\]

being

\[H = \frac{1}{2m}(P_x^2 + P_y^2) + V(x,y) + g|\psi(x,y)|^2 + \omega L_z\]

and \(\psi(t) = \psi_t(x,y)\) for the single component wave function, or

\[\begin{split}H = \begin{bmatrix} H_1 & \frac{\Omega}{2} \\ \frac{\Omega}{2} & H_2 \end{bmatrix}\end{split}\]

where

\[H_1 = \frac{1}{2m_1}(P_x^2 + P_y^2) + V_1(x,y) + g_1|\psi(x,y)_1|^2 + g_{12}|\psi(x,y)_2|^2 + \omega L_z\]\[H_2 = \frac{1}{2m_2}(P_x^2 + P_y^2) + V_2(x,y) + g_2|\psi(x,y)_2|^2 + g_{12}|\psi(x,y)_1|^2 + \omega L_z\]

and \(\psi(t) = \begin{bmatrix} \psi_1(t) \\ \psi_2(t) \end{bmatrix}\), for the two component wave function.

First we define the object for the external potential \(V(x,y)\). A general external potential function can be defined by a Python function, for instance, the harmonic potential can be defined as follows:

def harmonic_potential(x,y):
    return 0.5 * (x**2 + y**2)

Now we create the external potential object using the Potential class and then we initialize it with the function above:

potential = ts.Potential(grid)  # Create the potential object
potential.init_potential(harmonic_potential)  # Initialize it using a python function

Note that the module provides a quick way to define the harmonic potential, as it is fequently used:

omegax = omegay = 1.
harmonicpotential = ts.HarmonicPotential(grid, omegax, omegay)

We are ready to create the Hamiltonian object. For the sake of simplicity, let us create the Hamiltonian of the harmonic oscillator:

particle_mass = 1. # Mass of the particle
hamiltonian = ts.Hamiltonian(grid, potential, particle_mass)  # Create the Hamiltonian object

The quantum state is created by the State class; it resembles the way the potential is defined. Here we create the ground state of the harmonic oscillator:

import numpy as np  # Import the module numpy for the exponential and sqrt functions

def state_wave_function(x,y):  # Wave function
    return np.exp(-0.5*(x**2 + y**2)) / np.sqrt(np.pi)

state = ts.State(grid)  # Create the quantum state
state.init_state(state_wave_function)  # Initialize the state

The module provides several predefined quantum states as well. In this case, we could have used the GaussianState class:

omega = 1.
gaussianstate = ts.GaussianState(grid, omega)  # Create a quantum state whose wave function is Gaussian-like

We are left with the creation of the last object: the Solver class gathers all the objects we defined so far and it is used to perform the evolution and analyze the expectation values:

delta_t = 1e-3  # Physical time of a single iteration
solver = ts.Solver(grid, state, hamiltonian, delta_t)  # Creating the solver object

Finally we can perform both real-time and imaginary-time evolution using the method evolve:

iterations = 100  # Number of iterations to be performed
solver.evolve(iterations, True)  # Perform imaginary-time evolution
solver.evolve(iterations)  # Perform real-time evolution

Analysis

The classes we have seen so far implement several members useful to analyze the system (see the function reference section for a complete list).

Expectation values

The solver class provides members for the energy calculations. For instance, the total energy can be calculated using the get_total_energy member. We expect it to be \(1\) (\(\hbar =1\)), and indeed we get the right result up to a small error which depends on the lattice approximation:

tot_energy = solver.get_total_energy()
print(tot_energy)
1.00146456951

The expected values of the \(X\), \(Y\), \(P_x\), \(P_y\) operators are calculated using the members in the State class

mean_x = state.get_mean_x()  # Get the expected value of X operator
print(mean_x)
1.39431975344e-14

Norm of the state

The squared norm of the state can be calculated by means of both State and Solver classes

snorm = state.get_squared_norm()
print(snorm)
1.0

Particle density and Phase

Very often one is interested in the phase and particle density of the state. Two members of State class provide these features

density = state.get_particle_density()  # Return a numpy matrix of the particle density
phase = state.get_phase()  # Return a numpy matrix of the phase

Imprinting

The member imprint, in the State class, applies the following transformation to the state:

\[\psi(x,y) \rightarrow \psi'(x,y) = f(x,y) \psi(x,y)\]

being \(f(x,y)\) a general complex-valued function. This comes in handy when we want to imprint, for instance, vortices or solitons:

def vortex(x, y):  # Function defining a vortex
    z = x + 1j*y
    angle = np.angle(z)
    return np.exp(1j * angle)

state.imprint(vortex)  # Imprint the vortex on the state

File Input and Output

write_to_files and loadtxt members, in State class, provide a simple way to handle file I/O. The former writes the wave function arranged as a complex matrix, in a plain text; the latter loads the wave function from a file to the state object. The following code provides an example:

state.write_to_file("file_name")  # Write the wave function to a file
state2 = ts.State(grid)  # Create a new state
state2.loadtxt("file_name")  # Load the wave function from the file

For a complete list of methods see the function reference.