Nonclassicality of a photon-subtracted Gaussian field

We investigate the nonclassicality of a photon-subtracted Gaussian field, which was produced in a recent experiment, using negativity of the Wigner function and the nonexistence of well-behaved positive $P$ function. We obtain the condition to see negativity of the Wigner function for the case including the mixed Gaussian incoming field, the threshold photodetection and the inefficient homodyne measurement. We show how similar the photon-subtracted state is to a superposition of coherent states.

Figure 1

The fidelity between the photon-subtracted state and the ideal coherent-state superposition with an ideal single-photon detector (solid line) and a threshold detector (dotted line). The initial squeezing parameter is $\exp \left(2s\right)=2.36$ and the $x$ axis is the transmittivity of the beam splitter $T=t^2$. The amplitude $\alpha$ of the ideal coherent-state superposition is optimized for the maximum fidelity. The optimized amplitude $\alpha$ ranges between 1.02 (when $T=0.8$) and 1.16 (when $T\to 1$).

Figure 2

(Color online) (a) The Wigner function $W\left(z\right)$ of a photon-subtracted Gaussian state with a threshold detector for photon subtraction and ideal homodyne detectors for reconstruction of the Wigner function when $T=0.88$ and $\exp \left(2s\right)=2.36$. The minimum negativity is found as $W(0,0)=-0.52$. (b) The Wigner function of a photon-subtracted Gaussian state under the same condition as (a) but with homodyne efficiency $\eta =0.75$. The minimum negativity has been reduced to $-0.15$. (c) The Wigner function of a photon-subtracted Gaussian state under the same condition as (a) but with homodyne efficiency $\eta =0.75$ and with the modal purity factor 0.7. The negativity of the Wigner function has disappeared as $W\left(0\right)=0.075$.