# 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:

## Quadrature operators¶

The dimensionless **position and momentum quadrature operators** \(\x\) and \(\p\) are defined by

They fulfill the commutation relation

and satisfy the eigenvector equations

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:

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

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

## Number operator¶

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

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

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

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