# Operators¶

Note

In Strawberry Fields we use the convention \(\hbar=2\) by default, but other
conventions can also be chosen by setting the global variable `sf.hbar`

at the beginning of a session.
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 [1].

## References¶

- 1
F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders (eds.). NIST digital library of mathematical functions. Release 1.0.16 of 2017-09-18. [Online; accessed 2017-10-25]. URL: http://dlmf.nist.gov/.

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