Mathematical Optics

where we set ∂_j = ∂/∂x_j and x_j = x₊, x_z, x₋. For example, we obtain after some calculation

$⟨\text{μ};k|F(K_{+},K_z,K_{-})|\text{μ};k⟩=F^{(\mathrm{A})}({\mathrm{\partial }}_{-},{\mathrm{\partial }}_z,{\mathrm{\partial }}_{-}){\left.{\left[\frac{(1-|\text{μ}{|}^2)\exp (x_z/2)}{1-(\text{μ}+x_{+})({\text{μ}}^{\ast }+x_{-})\exp (x_z)}\right]}^{2k}\right|}_{x=0},$

(12.97)

for the Perelomov coherent state and

$⟨z;k|F(K_{+},K_z,K_{-})|z;k⟩=F^{(N)}(z^{\ast },{\mathrm{\partial }}_z,z){\left.\left[\frac{\exp (x_z/2)I_{2k-1}(2|z|\exp (x_z/2))}{I_{2k-1}(2|z|)}\right]\right|}_{x=0},$

(12.98)

for the Barut–Girardello coherent state

$⟨v;j|F(K_{+},K_z,K_{-})|v;j⟩=F^{(\mathrm{A})}({\mathrm{\partial }}_{-},{\mathrm{\partial }}_z,{\mathrm{\partial }}_{+}){\left.\left[\frac{1+(\text{μ}+x_{+})({\text{μ}}^{\ast }+x_{-})\exp (x_z)}{(1+|\text{μ}{|}^2)\exp (x_z/2)}\right]\right|}_{x=0},$

(12.99)

for the atomic coherent state.

Next, we consider average values in a quantum state described by the density operator ρ

$\text{ρ}=Z^{-1}\exp (a_{+}{K}_{+}+a_z{K}_z+a_{-}{K}_{-}),$

(12.100)

where Z = Tr exp(a₊K₊ + a_zK_z + a₋K₋) with Tr being the trace operation. The parameters a_± and a_z are required such that the density operator ρ