Quantum properties of exponential states

The use of Rényi entropy as an uncertainty measure alternative to variance leads to the study of states with quantum fluctuations below the levels established by Gaussian states, which are the position-momentum minimum uncertainty states according to variance. We examine the quantum properties of states with exponential wave functions, which combine reduced fluctuations with practical feasibility.

Figure 1

Wigner function $W_e(x,p)$ for the exponential state (3.10) for $⟨x⟩=⟨p⟩=0$ as a function of $x∕\gamma$ and $\gamma p$.

Figure 2

$d_B$ as a function of $\gamma$ for the exponential state (3.10). Minimum occurs around $\gamma \simeq 1.08$.

Figure 3

Number uncertainty ${\delta }_{e}n$ (solid line) as a function of $\gamma$ for exponential states with $\alpha =1$ and $⟨x⟩=⟨p⟩=0$, compared with the number uncertainty for a coherent state ${\delta }_{c}n$ (dashed line) with the same mean number ${⟨\widehat{n}⟩}_e={⟨\widehat{n}⟩}_c$.

Figure 4

Phase uncertainty in Eq. (7.3) where the shadowed region represents the area of uncertainty.

Figure 5

$\Delta x\Delta p$ and $\delta x\Delta p$ for the states (3.1) as functions of $\alpha$.