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Wigner function of cat state

  1. Aug 5, 2016 #1
    1. The problem statement, all variables and given/known data
    I am trying to calculate the wigner function for the even coherent state or cat state gives by,
    [tex] | \psi \rangle = N_+ \left( |\alpha \rangle + | -\alpha \rangle \right), [/tex]
    where ##|\alpha \rangle ## is a coherent state and ## |N_+|^2 = \dfrac{1}{ 2 + 2e^{-2|\alpha|^2}}##. I am doing this is a slightly round about manner, I am first calculating the Glauber-Sudarshan P distribution for the state and then calculating the Wigner function from there. I have obtained an expression but it is not agreeing with expressions I have found in reference books for the Wigner function of a even coherent state.
    2. Relevant equations
    To find the P distribution given a state ## \rho ##,

    [tex] P_{GS}(\gamma) = \frac{e^{|\gamma |^2}}{\pi^2} \int \langle -\beta | \rho | \beta \rangle e^{| \beta |^2}e^{\beta^* \gamma - \beta \gamma^*} d^2\beta[/tex]

    The Wigner function can be obtained from the P distrubtion as,
    [tex]W(\beta) = \frac{2}{\pi} \int P_{GS}(\gamma) e^{-2|\gamma - \beta|^2} d^2\gamma [/tex]

    Expected answer in real variables (such that ##\beta = q + ip ##) with ## \alpha ## real, from Introduction to quantum optics by Gerry & Knight,
    [tex] W(q, p) = \frac{1}{\pi (1 + \exp (-2\alpha^2))} \left\lbrace \exp (-2(q-\alpha)^2 -2p^2) + \exp (-2(q + \alpha)^2 -2p^2) \right. \\
    \left. + 2\exp (-2q^2 - 2p^2)\cos (4p\alpha)\right\rbrace [/tex]

    Inner product of coherent states is given by,

    [tex] \langle \alpha | \beta \rangle = \exp \left( -|\alpha|^2 / 2 - |\beta|^2 / 2 + \alpha^* \beta \right) [/tex]
    3. The attempt at a solution

    For us, ## \rho = |N_+|^2 ( | \alpha \rangle \langle \alpha | + | -\alpha \rangle \langle -\alpha | + | -\alpha \rangle \langle \alpha | + | \alpha \rangle \langle -\alpha |). ##
    [tex] P_{GS}(\gamma) = \sum\limits_{i = 1}^4 P_i ,[/tex]
    with,
    [tex] \begin{align*} P_1 &= \frac{|N_+|^2 e^{|\gamma |^2}}{\pi^2} \int \langle -\beta | \alpha \rangle \langle \alpha | \beta \rangle e^{| \beta |^2}e^{\beta^* \gamma - \beta \gamma^*} d^2\beta \\
    &= \frac{|N_+|^2 e^{|\gamma|^2 - |\alpha|^2 }}{\pi^2} \int \exp(\beta^*(\gamma - \alpha) - \beta (\gamma^* - \alpha^*)) d^2\beta \\
    &= |N_+|^2 e^{|\gamma|^2 - |\alpha|^2 } \delta^{(2)} ( \gamma - \alpha) \\
    &= |N_+|^2 \delta^{(2)} ( \gamma - \alpha) \end{align*} [/tex]

    ## P_2 ## the term corresponding to ## |-\alpha \rangle \langle -\alpha | ## can be obtained from ##P_1 ## by replacing ## \alpha ## with ## -\alpha ##, giving
    [tex] P_2 = |N_+|^2 \delta^{(2)} ( \gamma + \alpha) [/tex]

    For the cross term ## |-\alpha \rangle \langle \alpha | ##,
    [tex] \begin{align*} P_3 &= \frac{|N_+|^2 e^{|\gamma |^2}}{\pi^2} \int \langle -\beta | -\alpha \rangle \langle \alpha | \beta \rangle e^{| \beta |^2}e^{\beta^* \gamma - \beta \gamma^*} d^2\beta \\
    &= \frac{|N_+|^2 e^{|\gamma|^2 - |\alpha|^2 }}{\pi^2} \int \exp(\beta^*(\gamma + \alpha) - \beta (\gamma^* - \alpha^*)) d^2\beta \\
    &= \frac{|N_+|^2 e^{|\gamma|^2 - |\alpha|^2 }}{\pi^2} \int \exp (2\beta^*\alpha) \exp(\beta^*(\gamma - \alpha) - \beta (\gamma^* - \alpha^*)) d^2\beta \\
    &= \frac{|N_+|^2 e^{|\gamma|^2 - |\alpha|^2 }}{\pi^2} \int \exp (-2\alpha \frac{\partial}{\partial \alpha}) \exp(\beta^*(\gamma - \alpha) - \beta (\gamma^* - \alpha^*)) d^2\beta \\
    &= \frac{|N_+|^2 e^{|\gamma|^2 - |\alpha|^2 }}{\pi^2} \exp (-2\alpha \frac{\partial}{\partial \alpha}) \int \exp(\beta^*(\gamma - \alpha) - \beta (\gamma^* - \alpha^*)) d^2\beta \\
    &= |N_+|^2 e^{|\gamma|^2 - |\alpha|^2 } \exp (-2\alpha \frac{\partial}{\partial \alpha})\delta^{(2)} ( \gamma - \alpha) \\
    &= |N_+|^2 \exp (-2\alpha \frac{\partial}{\partial \alpha})\delta^{(2)} ( \gamma - \alpha) \end{align*} [/tex]
    Again, ## P_4 ## can be obtained from ## P_3 ## by replacing ## \alpha## by ## -\alpha ##.
    [tex] P_4 = |N_+|^2 \exp (-2\alpha \frac{\partial}{\partial \alpha})\delta^{(2)} ( \gamma + \alpha) [/tex]
    Finally putting the 4 terms together the P distribution is given by,

    [tex] P_{GS}(\gamma) = |N_+|^2 \left\lbrace 1 + \exp ( -2\alpha\frac{\partial}{\partial \alpha}) \right\rbrace \left[ \delta^{(2)} ( \gamma - \alpha) + \delta^{(2)} ( \gamma + \alpha) \right] [/tex]

    The wigner function can be obtained using the relation given in the begining as,
    [tex] W(\beta) = \frac{2 |N_+|^2}{\pi} \left\lbrace1 + \exp(-2\alpha\frac{\partial}{\partial \alpha}) \right\rbrace \left[ e^{-2|\alpha - \beta|^2} + e^{-2|\alpha + \beta|^2} \right][/tex]

    Now,
    [tex] \exp(-2\alpha\frac{\partial}{\partial \alpha}) e^{-2|\alpha - \beta|^2} = \exp (4(\alpha^* - \beta^*)\alpha)e^{-2(\alpha - \beta)(\alpha^* - \beta^*)} = e^{2(\alpha + \beta )(\alpha^* - \beta^*)} [/tex]
    replacing ##\alpha ## by ## - \alpha ## again gives,

    [tex] \exp(-2\alpha\frac{\partial}{\partial \alpha}) e^{-2|\alpha + \beta|^2} = e^{2(\alpha - \beta )(\alpha^* + \beta^*)} [/tex]

    Now the wigner function can be written as,

    [tex] W(\beta) = \frac{2 |N_+|^2}{\pi} \left[ e^{-2|\alpha - \beta|^2} + e^{-2|\alpha + \beta|^2} + e^{2(\alpha + \beta )(\alpha^* - \beta^*)} + e^{2(\alpha - \beta )(\alpha^* + \beta^*)} \right] [/tex]

    I will now write this function in terms of real variables to compare with the answer I already have. Taking,
    ## \alpha = q_0 + ip_0 ## and ##\beta = q + ip ##, and substituting for #|N_+|^2#. Note also that the last two terms are complex conjugates of each other and so can be written as twice the real part of each.
    [tex] W(q, p) = \frac{1}{\pi (1 + \exp( -2(q_0^2 + p_0^2 ) ) } \left[ e^{-2(q - q_0)^2}e^{-2(p - p_0)^2} + e^{-2(q + q_0)^2}e^{-2(p + p_0)^2} \\ + 2\Re ( e^{2 \lbrace (q_0 - q) + i(p_0 - p) \rbrace \lbrace (q_0 + q ) +i(p_0 + p) \rbrace }) \right] [/tex]

    Simplifying the last term gives,

    [tex] W(q, p) = \frac{1}{\pi (1 + \exp( -2(q_0^2 + p_0^2 ) ) } \left[ e^{-2(q - q_0)^2}e^{-2(p - p_0)^2} + e^{-2(q + q_0)^2}e^{-2(p + p_0)^2} \\ + 2 e^{2 [(q_0^2 + p_0^2) - (q^2 + p^2) ] }\cos 4(p_0q - q_0p) \right] [/tex]

    Now to reproduce the result I have, I should substitute, ## q_0 = \alpha ## and ## p_0 = 0 ##
    This gives,

    [tex] W(q, p) = \frac{1}{\pi (1 + \exp( -2 \alpha^2 ) } \left[ e^{-2(q - \alpha)^2}e^{-2p^2} + e^{-2(q + \alpha)^2}e^{-2p^2} \\ + 2 e^{2 [\alpha^2 - (q^2 + p^2) ] }\cos(4\alpha p) \right] [/tex]

    The expected answer is,

    [tex] W(q, p) = \frac{1}{\pi (1 + \exp (-2\alpha^2))} \left\lbrace \exp (-2(q-\alpha)^2 -2p^2) + \exp (-2(q + \alpha)^2 -2p^2) \right. \\
    \left. + 2\exp (-2q^2 - 2p^2)\cos (4p\alpha)\right\rbrace [/tex]

    The 3rd term doesn't agree to a factor of ## e^{2\alpha^2}##. I am unable to find my error, any help would be appreciated :-).
     
  2. jcsd
  3. Aug 10, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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