Adding or Subtracting a single Photon is the same for Pure Squeezed Vacuum States

Abstract

The addition of a single photon to a light field can lead to exactly the same \emph{outcome} as the subtraction of a single photon. We prove that this \cterm is true for pure squeezed vacuum states of light, and in some sense only for those. We show that mixed states can show this \cterm for addition or subtraction of a photon if they are generated from incoherent sums of pure squeezed vacuum states with the same squeezing. We point out that our results give a reinterpretation to the fact that pure squeezed vacuum states, with squeezing $e^{-z}$, are formally annihilated by Bogoliubov-transformed annihilation operators: $\hat a_z = \hat a \cosh(z) - \hat a^\dagger \sinh(z) $.

Figures (3)

• Figure 1: Using a squeezed vacuum $W(x,p,0,\sigma_x \! = \! 4,\sigma_p \! = \! 1/2)$ input state (\ref{['Eq:ImPureSqueezedVacuumState_W']}) that is impure: Plots of the renormalized outcome states after a photon is added [$W_+$ of Eq. (\ref{['Eq:ImPureSqueezedVacuumState_W_Add']}), left panel] or subtracted [$W_-$ of Eq. (\ref{['Eq:ImPureSqueezedVacuumState_W_Sub']}), middle panel] show that their difference is non-zero [$\Delta W = W_+ - W_- \neq 0$, right panel]. The impurity violates the identity-of-outcome condition (\ref{['eq:_aWad_adWa']}).
• Figure 2: Plot of photon-added or -subtracted cases for input states $W_{\rm II}$ (\ref{['Eq:W_II']}), with parameters $P=0.5$, $\sigma_x=2.2$, and orienta-tion angles $\theta_1 = 0$ and $\theta_2 = \frac{\pi}{4}$ (left panel) and maximally mixed state $W_{[2 \pi ]} (\sigma_x)$ (\ref{['Eq:W_2Pi']}), with $\sigma_x=2.2$ (right panel). In both cases $W[0,0]=-\frac{1}{\pi}$ and both fulfil condition (\ref{['eq:_aWad_adWa']}).
• Figure 3: Radial profile of maximally mixed state (\ref{['Eq:W_2Pi']}) log$_{10}[W_{[2 \pi ]} (x, 0, \sigma_x)]$, with fixed $\sigma_x=2.2$ (left panel, red curve; for contrast, the black dotted curve shows a gaussian profile) demonstrating clear deviation from a gaussian profile. Purity of maximally mixed state $\cal P$$(W_{[2 \pi ]} (\sigma_x))$ as a function of squeezing $\sigma_x$ (right panel).