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Idoubt
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Homework Statement
I am trying to calculate the wigner function for the even coherent state or cat state gives by,
| \psi \rangle = N_+ \left( |\alpha \rangle + | -\alpha \rangle \right),
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.
Homework Equations
To find the P distribution given a state ## \rho ##,
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
The Wigner function can be obtained from the P distrubtion as,
W(\beta) = \frac{2}{\pi} \int P_{GS}(\gamma) e^{-2|\gamma - \beta|^2} d^2\gamma
Expected answer in real variables (such that ##\beta = q + ip ##) with ## \alpha ## real, from Introduction to quantum optics by Gerry & Knight,
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. \\ <br /> \left. + 2\exp (-2q^2 - 2p^2)\cos (4p\alpha)\right\rbrace
Inner product of coherent states is given by,
\langle \alpha | \beta \rangle = \exp \left( -|\alpha|^2 / 2 - |\beta|^2 / 2 + \alpha^* \beta \right)
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 |). ##
P_{GS}(\gamma) = \sum\limits_{i = 1}^4 P_i ,
with,
\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 \\<br /> &= \frac{|N_+|^2 e^{|\gamma|^2 - |\alpha|^2 }}{\pi^2} \int \exp(\beta^*(\gamma - \alpha) - \beta (\gamma^* - \alpha^*)) d^2\beta \\<br /> &= |N_+|^2 e^{|\gamma|^2 - |\alpha|^2 } \delta^{(2)} ( \gamma - \alpha) \\<br /> &= |N_+|^2 \delta^{(2)} ( \gamma - \alpha) \end{align*}
## P_2 ## the term corresponding to ## |-\alpha \rangle \langle -\alpha | ## can be obtained from ##P_1 ## by replacing ## \alpha ## with ## -\alpha ##, giving
P_2 = |N_+|^2 \delta^{(2)} ( \gamma + \alpha)
For the cross term ## |-\alpha \rangle \langle \alpha | ##,
\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 \\ <br /> &= \frac{|N_+|^2 e^{|\gamma|^2 - |\alpha|^2 }}{\pi^2} \int \exp(\beta^*(\gamma + \alpha) - \beta (\gamma^* - \alpha^*)) d^2\beta \\ <br /> &= \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 \\ <br /> &= \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 \\<br /> &= \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 \\<br /> &= |N_+|^2 e^{|\gamma|^2 - |\alpha|^2 } \exp (-2\alpha \frac{\partial}{\partial \alpha})\delta^{(2)} ( \gamma - \alpha) \\ <br /> &= |N_+|^2 \exp (-2\alpha \frac{\partial}{\partial \alpha})\delta^{(2)} ( \gamma - \alpha) \end{align*}
Again, ## P_4 ## can be obtained from ## P_3 ## by replacing ## \alpha## by ## -\alpha ##.
P_4 = |N_+|^2 \exp (-2\alpha \frac{\partial}{\partial \alpha})\delta^{(2)} ( \gamma + \alpha)
Finally putting the 4 terms together the P distribution is given by,
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]
The wigner function can be obtained using the relation given in the beginning as,
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]
Now,
\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^*)}
replacing ##\alpha ## by ## - \alpha ## again gives,
\exp(-2\alpha\frac{\partial}{\partial \alpha}) e^{-2|\alpha + \beta|^2} = e^{2(\alpha - \beta )(\alpha^* + \beta^*)}
Now the wigner function can be written as,
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]
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.
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]
Simplifying the last term gives,
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]
Now to reproduce the result I have, I should substitute, ## q_0 = \alpha ## and ## p_0 = 0 ##
This gives,
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]
The expected answer is,
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. \\ <br /> \left. + 2\exp (-2q^2 - 2p^2)\cos (4p\alpha)\right\rbrace
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 :-).