$W$-representation

An alternative phase-space method is to convert the operator master equation into a PDE for the Wigner function $W(A;z)$ of an operator $A$. We recall Formula (4.177b) for the Wigner function of an operator product $AB$,

$$W(AB;z)= W(A;z) \exp \left[ \frac{1}{2}\left(\overleftarrow{\part... ...erleftarrow{\partial}_{z^*} \overrightarrow{\partial}_{z} \right)\right] W(B;z)$$ (95)

We obtain

$$W(a) = z,\quad W(a^{\dagger})=z^*$$

$$W(a^{\dagger}a) = z^* \left(1+\frac{1}{2}\left(\partial_z\partial_{z^*}-\partial_{z^*}\partial_{z}\right)\right) z =z^*z -\frac{1}{2}$$

$$W(a^{\dagger}a\rho) = \left(z^*z -\frac{1}{2}\right) \left(1+\frac{1}{2}\left(\partial_... ...\left(\partial_z\partial_{z^*}-\partial_{z^*}\partial_{z}\right) \right)W(\rho)$$

$$= \left(z^*z -\frac{1}{2}\right)W(\rho) +\frac{1}{2}z^*\partial_{z^... ...\frac{1}{2}z\partial_{z} W(\rho) -\frac{2}{8}\partial_{z}\partial_{z^*} W(\rho)$$

$$W(\rho a^{\dagger}a) = W(\rho)\left(1+\frac{1}{2}\left(\partial_z\partial_{z^*}-\partial... ...l_{z^*}-\partial_{z^*}\partial_{z}\right) \right)\left(z^*z -\frac{1}{2}\right)$$

$$= \left(z^*z -\frac{1}{2}\right)W(\rho) -\frac{1}{2}z^*\partial_{z^... ...\frac{1}{2}z\partial_{z} W(\rho) -\frac{2}{8}\partial_{z}\partial_{z^*} W(\rho)$$ (96)

Similarly,

$$W(a\rho) = z \left(1+\frac{1}{2}\left(\partial_z\partial_{z^*}-\partial_{z^*}\partial_{z}\right)\right) W(\rho) = zW(\rho) +\frac{1}{2}\partial_{z^*}W(\rho)$$

$$W(a\rho a^{\dagger}) = \left(zW(\rho) +\frac{1}{2}\partial_{z^*}W(\rho)\right) \left(1+\... ...}{2}\left(\partial_z\partial_{z^*}-\partial_{z^*}\partial_{z}\right)\right) z^*$$

$$= \left(zz^*W(\rho) +\frac{1}{2}z^*\partial_{z^*}W(\rho)\right) +\frac{1}{2}\partial_z(zW(\rho))+\frac{1}{4}\partial_z\partial_{z^*}W(\rho)$$

$$W(a^{\dagger}\rho) = z^* \left(1+\frac{1}{2}\left(\partial_z\partial_{z^*}-\partial_{z^*}\partial_{z}\right)\right) W(\rho) = z^*W(\rho) -\frac{1}{2}\partial_{z}W(\rho)$$

$$W(a^{\dagger}\rho a) = \left(z^*W(\rho) -\frac{1}{2}\partial_{z}W(\rho)\right) \left(1+\frac{1}{2}\left(\partial_z\partial_{z^*}-\partial_{z^*}\partial_{z}\right)\right)z$$

$$= \left(zz^*W(\rho) -\frac{1}{2}z\partial_{z}W(\rho)\right) -\frac{1}{2}\partial_{z^*}(z^*W(\rho))+\frac{1}{4}\partial_z\partial_{z^*}W(\rho)$$

Thus,

$$\Big\{ a^{\dagger} a {\rho} + \rho a^{\dagger} a - 2 a \rho a^{\dagger} \Big\} \leftrightarrow -\Big\{2+z\partial_z+z^*\partial_{z^*}+\partial_z\partial_{z^*}\Big\}W(\rho)$$

$$\Big\{ a^{\dagger} a {\rho} +\rho (a^{\dagger}a+1) - a \rho a^{\dagger} -a^{\dagger} \rho a \Big\} \leftrightarrow -\partial_z\partial_{z^*}W(\rho)$$

$$\left[ a^{\dagger} a,\rho \right] \leftrightarrow \left(z^*\partial_{z^*}-z\partial_z\right)W(\rho).$$ (97)

Therefore, the master equation Eq.(7.69) is converted into

$$\frac{\partial}{\partial t}W(z,t) = -i\bar{\Omega} \left(z^*\partial_{z^*}-z\partial_z\right)W(z,t) +\kappa\Big\{2+z\partial_z+z^*\partial_{z^*}+\partial_z\partial_{z^*}\Big\}W(z,t)$$

$$+ 2\kappa n_B(\Omega)\partial_z\partial_{z^*}W(z,t)$$

$$=\left\{ 2\kappa \right. + \left. i\left[ \bar{\Omega} - i\kappa\right]z\frac{\partial}{\par... ...z^*} +\kappa[1+2 n_B] \frac{\partial^2}{\partial z^*\partial z}\right\} W(z,t).$$

We compare this with the PDE for the $P$-function, Eq.(7.79):

$$\frac{\partial}{\partial t}P(z,t) = \left\{ 2\kappa + i\left[ \ba... ...tial z^*} +2\kappa n_B \frac{\partial^2}{\partial z^*\partial z}\right\} P(z,t)$$

The difference is just in the diffusion term, i.e., $1+2 n_B$ in the Wigner representation instead of $2 n_B$ in the P representation. In the Wigner representation, even at zero temperature $T=0$ ($n_B=0$) one has a diffusion term in the PDE. Technically, the solution proceeds as before: one first solves the first order part via characteristics and then the diffusive part via Fourier transformation.

• A similar derivation can be done for the $Q$-representation, cf. Walls/Milburn. The $Q$-representation is more convenient for systems where the initial oscillator state is squeezed, or the decay is into a bath not in thermal equilibrium but in a squeezed state.