# Operators¶

Note

In the Strawberry Fields we use the convention $$\hbar=2$$ by default, but other conventions can also be chosen on engine initialization. In this document we keep $$\hbar$$ explicit.

## Annihilation and creation operators¶

As noted in the introduction, some of the basic operators used in CV quantum computation are the bosonic anhilation and creation operators $$\a$$ and $$\ad$$. The operators corresponding to two seperate modes, $$\a_1$$ and $$\a_2$$ respectively, satisfy the following commutation relations:

$\begin{split}&[\a_1,\ad_1] = [\a_2,\ad_2] = \I,\\ &[\a_1,\a_1]=[\a_1,\ad_2]=[\a_1,\a_2]=[\a_2,\a_2]=0.\end{split}$

The dimensionless position and momentum quadrature operators $$\x$$ and $$\p$$ are defined by

$\x = \sqrt{\frac{\hbar}{2}}(\a+\ad),~~~ \p = -i \sqrt{\frac{\hbar}{2}}(\a-\ad).$

They fulfill the commutation relation

$[\x, \p] = i \hbar,$

and satisfy the eigenvector equations

$\x\xket{x} = x \xket{x} ~~~~ \p\ket{p}_p = p\ket{p}_p$

Here, $$\xket{x}$$ and $$\ket{p}_p$$ are the eigenstates of $$\x$$ and $$\p$$ with eigenvalues $$x$$ and $$p$$ respectively. The position and momentum operators generate shifts in each others’ eigenstates:

$\begin{split}e^{-i r \p/\hbar} \xket{x} = \xket{x+r},\\ e^{i s \x/\hbar} \ket{p}_p = \ket{p+s}_p.\end{split}$

In the vacuum state, the variances of position and momentum are given by

$\bra{0}\x^2\ket{0} = \bra{0}\p^2\ket{0} = \frac{\hbar}{2}.$

Note that we can also write the annihilation and creation operators in terms of the quadrature operators:

$\a := \sqrt{\frac{1}{2 \hbar}} (\x +i\p), ~~~~ \ad := \sqrt{\frac{1}{2 \hbar}} (\x -i\p).$

## Number operator¶

The number operator is $$\hat{n} := \ad \a$$, and satisfies the eigenvector equation

$\hat{n}\ket{n} = n\ket{n}$

where $$\ket{n}$$ are the Fock states with eigenvalue $$n$$. Furthermore, note that

$\ad\ket{n} = \sqrt{n+1}\ket{n+1}~~~\text{and}~~~~\a\ket{n}=\sqrt{n}\ket{n-1}.$

Using the position eigenstates $$\xket{x}$$, we may represent the Fock states $$\ket{n}$$ as wavefunctions in the position representation:

$\psi_n(x) = \braket{x_x|n} = \frac{1}{\sqrt{2^n\,n!}} \: \left(\frac{1}{\pi \hbar}\right)^{1/4} e^{- \frac{1}{2 \hbar}x^2} \: H_n\left(x/\sqrt{\hbar} \right), \qquad n = 0,1,2,\ldots,$

where $$H_n(x)$$ are the physicist’s Hermite polynomials [45].