How squeezed states both maximize and minimize the same notion of quantumness

Beam splitters are routinely used for generating entanglement between modes in the optical and microwave domains, requiring input states that are not convex combinations of coherent states. This leads to the ability to generate entanglement at a beam splitter as a notion of quantumness. A similar, yet distinct, notion of quantumness is the amount of entanglement generated by two-mode squeezers (i.e., four-wave mixers). We show that squeezed-vacuum states, paradoxically, both minimize and maximize these notions of quantumness, with the crucial resolution of the paradox hinging upon the relative phases between the input states and the devices. Our notion of quantumness is intrinsically related to eigenvalue equations involving creation and annihilation operators, governed by a set of inequalities that leads to generalized cat and squeezed-vacuum states.

Figure 1

Wigner quasiprobability distributions for higher-order cat states with $\alpha =5$ and $n=2$, 5, and 8. These states saturate the inequality of Eq. (27) and can be considered to maximize that form of quantumness. The $x$ and $y$ axes correspond to the dimensionless $x$ and $p$ quadratures of a harmonic oscillator, such as an electromagnetic field, in units where $[\widehat{x},\widehat{p}]=i$. The Wigner functions are large near the values $\alpha e^{2\pi ik/n}$ and display their rotational symmetries. They also display their oscillatory structure wherein different coherent states interfere and the Wigner functions take negative values. All figures were created using QuTiP [86, 87].

Figure 2

Wigner quasiprobability distributions for the generalized states satisfying the eigenvalue equation of Eq. (2) and saturating the inequality of Eq. (31) for values of $k$ and $l$ ranging from 1 to 4 and various values of $\eta$ (0.2, 1.0, and 5.0 for $k>l$ and 0.2, 0.5, and 1.0 for $k=l$). The states display squeezing from $n=k+l$ directions, with the strength of the squeezing increasing with $\left|\eta \right|$. Changing the phase of $\eta$ will rigidly rotate these distributions about the origin. Some negative values of the Wigner function are visible at the $n$ locations where the squeezing occurs, becoming more negative with increasing $\left|\eta \right|$. Subplots (c) and (l) cover a larger area than the rest to fully showcase all of the relevant phase-space structures.