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Root of number operator?

  1. May 1, 2009 #1

    KFC

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    I seen in some paper, there is an matrix whose element has a square root of number operator, e.g.

    [tex]
    A = \left(
    \begin{matrix}
    \alpha & \gamma \sqrt{\hat{a}\hat{a}^\dagger} \\
    -\gamma \sqrt{\hat{a}^\dagger\hat{a} & \beta
    \end{matrix}
    \right)
    [/tex]
    where [tex]\alpha, \beta, \gamma[/tex] are real number.

    What is [tex]A^\dagger[/tex]? Can I write it as the following?
    [tex]
    A^\dagger = \left(
    \begin{matrix}
    \alpha & -\gamma \sqrt{\hat{a}^\dagger\hat{a}} \\
    \gamma \sqrt{\hat{a}\hat{a}^\dagger & \beta
    \end{matrix}
    \right)
    [/tex]

    By the way, if I have it operate on any Fock state, how could the operators in the matrix operating those states?
     
  2. jcsd
  3. May 1, 2009 #2
    The square root of the number operator is probably defined by it's taylor expansion. This means that letting the the root of the number operator act on a Fock-state gives you [itex]\sqrt{N}[/itex] times the state. Also since the number operator is self-adjoint this implies that any function of it will be self-adjoint, in particular [itex](\sqrt{a^\dag a})^\dag=\sqrt{a^\dag a}[/itex].

    One can think of the number operator as matrices themselves (and thus also the square root of the number operator) so what you have is a 2x2 block-matrix. The hermitian adjoint of which is given by what you wrote.

    I don't think that this matrix can act on the Fock-space generated by the algebra of a^dag and a. I could imagine that the matrix never acts on the Fock-space by itself only in the form of some "scalar product". What I mean with this is basically some new operator C:

    [tex]
    C=(a_1, a_2)A\begin{pmatrix}b_1\\ b_2\end{pmatrix}
    [/tex]

    where a_1/2 and b_1/2 can be numbers or operators. Then C can act on the Fock-space.

    Compare it with how one sometimes uses "vectors" of operators like

    [tex]
    c=\begin{pmatrix}
    a_1 \\
    a_2
    \end{pmatrix}, \quad c^\dag=(a_1^\dag, a_2^\dag)
    [/tex]

    but they never really act on Fock-space in this vector form but only in a "scalar product" form. For example the Hamiltonian may look something like

    [tex]\mathcal{H}=c^\dag H c[/tex]

    usually the elements of the matrix H are real numbers but in principle they could also be operators.
     
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