Quadratic Hamiltonian Diagonalization

A project which aims at diagonalizing a Quadratic (Bosonic) Hamiltonian using known methods from literature.

The main goal is to obtain the energies (excited spectra) for the quadratic Hamiltonian (of bosonic type).

The quadratic Hamiltonian under study is of bosonic type (written in terms of bosonic creation and annihilation operators $b^\dagger$ and $b$, respectively). The harmonic term is parametrized by a real constant $\epsilon$ and the quadratic terms depend on the real parameter $v$. The actual expression of $H$ is given below: $$H=\epsilon b^\dagger b -v(b^\dagger b^\dagger+bb)$$

Main goals of this project:

• Use the occupation number representation to diagonalize the Hamiltonian.
• Diagonalization of the Hamiltonian basically implies to get $H$ under a matrix form, then find its eigenvalues and eigenvectors.
• The eigenvalues of $\mathcal{M}(H)$ will represent the possible states that the system can have.
• Since the representation is given by the bosonic number states $|0\rangle$, $|1\rangle$, ..., $|n\rangle$, with $n$ arbitrarily large, the matrix will have a size of $n\times n$.
• This present formalism is based on a numerical approach, so there must be a truncation order $\eta$ for fixing the size of the matrix before implementing the diagonalization procedure.
• Once $\eta$ has been fixed, the bosonic states up to $|\eta\rangle$ will be used as basis for getting the matrix elements of $H$ (denoted by $H_{nm}$, with $n,m$ representing the actual number states).
• Having the basis ready, and knowing how the bosonic creation and annihilation operators act on these states, the procedure for finding the eigenvalues will be straightforward:

Solving the characteristic equation will give the eigenvalues of $H$

$$det[\mathcal{M}(H)-\lambda I]=0$$

The set $$\Lambda=[\lambda_1,\dots,\lambda_n]$$ will be the set of eigenstates of $H$.

The eigenvectors for the solutions verify the equation: $$\mathcal{M}x_i=\lambda_i x_i$$

Having $$\Lambda$$ for a given $H$ (parametrized in terms of $\mathcal{P}={\epsilon,v})$ means that the solutions have been found and the system has defined physical states. Thus, problem is solved.

Implementations

The process of diagonalization is done in:

1. Mathematica (see Wolfram Mathematica .nb notebooks in the project tree)
2. Python (see python scripts attached in the project tree)
3. (?optional) Javascript

Graphical representation of $\lambda$ in terms of q-ratio

The two parameters that define the Hamiltonian are encoded into a single parameter called q-ratio ($q\equiv\frac{v}{\epsilon}$).

Every solution of the system will be a function of this ratio, so the values of $\Lambda$ will change with the change in $q$. In the images below, one can see the evolution of the 6th and 10th solutions of the system (with truncation order N=10).

September 2020 update

This page describes in detail how the numerical implementation works. The latest updates fixed an inconsistency in the eigenvalue behavior with respect to the $q$-parameter. Every $\lambda$ which belongs to the physical solutions of the Hamiltonian $H$ (with $N$-truncation order and $q$ parameter) has a well-defined evolution, and both Python as well as Mathematica implementations can graphically represent each $\lambda_i=f(q)$ with $i=1,\dots,N$.

Graphical representation of the eigenvalues of $H$

This example shown below is for a Hamiltonian with $N=10$ truncation order (which means that the first 10 even and odd, respectively, states are considered in the matrix associated to $H$).

The evolution of $\lambda_5$, the fifth solution of $H(10)$ is evaluated for the interval $\in[0,3]$. Both the even case as well as the odd case are shown.

Grid Plot implementation with Py3

This implementation is used for a graphical representation (in a batch-manner) of the eigenvalues of the boson Hamiltonian.

[x] - Update the python implementation with a grid plot.

An example of grid plot can be seen below. Core implementation has been done. Source

Actual grid plot implementation can be seen in this commit. The data structure to be used for importing into the plot function has the following structure: