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:

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

Quadrature operators

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 [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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