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Problem with Hermicity of Density matrix

  1. May 1, 2012 #1
    Hi, I am trying to solve a modified Jayne's cummings model using the Von Neumann equation and Wigner function but am having a problem with Density matrix hermicity;

    I am trying to solve in Schrodinger picture.

    I have my system Hamiltonian as;

    [tex]
    H_{0} = \frac{1}{2}\hbar \Omega \sigma_{z} + \hbar \omega aa^{\dagger}
    [/tex]


    I am having a problem with the second term of this system Hamiltonian...when I follow it through into the Von neumann equation and Wigner function I get a non-Hermitian matrix??

    I can't find a problem with the maths...the only thing I can think is to drop the second term and leave it out of the Von Neumann equation...if this is possible, why?

    Thanks.
     
  2. jcsd
  3. May 1, 2012 #2

    vanhees71

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    2016 Award

    In the Schroedinger picture of time evolution the full time dependence is on the states, and thus the (picture-independent) von Neumann equation, translates into

    [tex]\frac{\mathrm{d}}{\mathrm{d} t} \hat{R}=\frac{1}{\mathrm{i} \hbar} [\hat{R},\hat{H}]+ \left (\frac{\partial \hat{R}}{\partial t} \right )_{\text{expl}}.[/tex]

    Since [itex]\hat{R}[/itex] is Hermitian at the initial time and [itex]\hat{H}[/itex] is Hermitian, [itex]\hat{R}[/itex] is Hermitian at any time.

    So there must be some mistake in your calculation, because your Hamiltonian is obviously Hermitian. To find, what's wrong, you should post your concrete problem and your ansatzes for the solution.
     
  4. May 18, 2012 #3
    Thanks, this is my problem;

    [tex]

    H=H_{0} + H_{I}

    [/tex]

    Where

    [tex]
    H_{0}= \frac{1}{2} \hbar \Omega \sigma_{z} + \hbar \omega a^{\dagger}a
    [/tex]

    and

    [tex]
    H_{I}=g_{1} \sigma_{x} (a+a^{\dagger}) + g_{2} \sigma_{z} (a+a^{\dagger}) +g_{3} \sigma_{z} (a^{2} + aa^{\dagger} + a^{\dagger}a +a^{\dagger 2})
    [/tex]

    I put this into the von neumann equation;

    [tex]
    i\hbar \frac{\partial}{\partial t} \rho = \left[ H_{0} + H_{I} , \rho \right]
    [/tex]

    Using the density matrix as a 2x2 matrix with elements [itex] \rho_{11} , \rho_{12}, \rho_{21}, \rho_{22} [/itex].

    I then put all the equations into the von Neumann equation and multiply out all of the matrices and split the density matrix into its component parts. I then convert this from being a function of a and [itex] a^{\dagger} [/itex] into the Wigner function of x and y (using standard conversion and correspondences). The Wigner function should still be Hermitian. Below are the four matrix elements of the Wigner function;

    [tex]
    i \hbar \frac{\partial}{\partial t} W_{11} = \hbar \omega \left(-1 + ix \frac{\partial}{\partial y} - iy \frac{\partial}{\partial x} \right) W_{11}
    +g_{1} \left( 2x + \frac{1}{2} i \frac{\partial}{\partial y} \right) W_{21}
    -g_{1} \left( 2x - \frac{1}{2} i \frac{\partial}{\partial y} \right) W_{12}
    +g_{2} i \frac{\partial}{\partial y} W_{11}
    +g_{3} \left( 4ix \frac{\partial}{\partial y} \right) W_{11}
    [/tex]

    [tex]
    i \hbar \frac{\partial}{\partial t } W_{12} = \hbar \Omega W_{12} + \hbar \omega \left( -1 + ix \frac{\partial}{\partial y} - iy \frac{\partial}{\partial x} \right) W_{12}
    +g_{1} \left( 2x + \frac{1}{2} i \frac{\partial}{\partial y} \right) W_{22}
    -g_{1} \left( 2x - \frac{1}{2} i \frac{\partial}{\partial y} \right) W_{11}
    +4g_{2}xW_{12}
    +g_{3} \left( 8x^{2} - \frac{1}{2} \frac{\partial^{2}}{\partial y^{2}} \right) W_{12}
    [/tex]

    [tex]
    i \hbar \frac{\partial}{\partial t } W_{21} = - \hbar \Omega W_{21} + \hbar \omega \left( -1 + ix \frac{\partial}{\partial y} - iy \frac{\partial}{\partial x} \right) W_{21}
    +g_{1} \left( 2x + \frac{1}{2} i \frac{\partial}{\partial y} \right) W_{11}
    -g_{1} \left( 2x - \frac{1}{2} i \frac{\partial}{\partial y} \right) W_{22}
    -4g_{2}xW_{21}
    -g_{3} \left( 8x^{2} - \frac{1}{2} \frac{\partial^{2}}{\partial y^{2}} \right) W_{21}
    [/tex]


    [tex]
    i \hbar \frac{\partial}{\partial t} W_{22} = \hbar \omega \left(-1 + ix \frac{\partial}{\partial y} - iy \frac{\partial}{\partial x} \right) W_{22}
    +g_{1} \left( 2x + \frac{1}{2} i \frac{\partial}{\partial y} \right) W_{12}
    -g_{1} \left( 2x - \frac{1}{2} i \frac{\partial}{\partial y} \right) W_{21}
    -g_{2} i \frac{\partial}{\partial y} W_{22}
    -g_{3} \left( 4ix \frac{\partial}{\partial y} \right) W_{22}
    [/tex]

    Which are non-Hermitian. I have checked the calculation several times and cannot find a mathematical error. I can only think I've omitted something. For example, do I need to apply some unitary operator somewhere, eg;

    [tex]
    U(t,t_{0})= e^{i \hbar H_{0}(t_0)}
    [/tex]

    Am I missing something?

    Thanks for any help, is much appreciated.
     
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