Squeezed quantum multiplets: properties and phase space representation

TL;DR

This work defines squeezed multiplets as rotationally invariant, D-element encodings built from superpositions of D squeezed states, and extends the construction to higher-order squeezing. It provides a complete analytic phase-space description for ordinary multiplets via the Wigner and characteristic functions, and presents explicit expressions showing how zeros and oscillations in the characteristic function encode hypersensitivity to perturbations for metrological applications. The paper also compares ordinary and higher-order multiplets, showing that overlaps can be large only in limited parameter regimes and that higher-order states demand numerical treatment, while both offer error-detectable encoding through parity-based measurements. The results establish a framework for phase-space analysis of non-Gaussian, rotat ionally symmetric CV codes with potential experimental realizations and metrological uses, including asymptotic insights into the zeros’ geometry at large squeezing and high multiplet dimension.

Abstract

We define and study the properties of ``squeezed quantum multiplets''. Ordinary multiplets are sets of $D$-orthonormal quantum states formed by superpositions of states squeezed along $D$ equally spaced directions in quadrature space. More generally, we also discuss superpositions of ``higher-order squeezed states'', including tri-squeezed and quad-squeezed states. All these states involve superpositions of multiples of $p$ photons. We compare states in ordinary ($p=2$) multiplets and higher-order ones ($p>2$) in the most relevant cases, showing that ordinary squeezed multiplets and higher-order ones share some important similarities, as well as some differences. Finally, we present analytical expressions for phase-space distributions (Wigner and characteristic functions) representing ordinary squeezed multiplets. We use this to show that some squeezed multiplets are highly sensitive to perturbations in all phase-space directions, making them interesting for metrological applications.

Figures (5)

• Figure 1: Overlap between (a) trisqueezed states, (b) even, and (c) odd quadsqueezed states with their corresponding multiplets $D=3$, $m=0$; $D=4$, $m=0$; and $m=2$, respectively. The overlap is represented by a color scale as a function of the squeezing ($r$) and $p$th-order squeezing ($r_p$) parameters, with black crosses indicating the maximum value for each $r_p$. Note that the color scale in (a) has a different slope from $0.85$ to $0.99$ than from $0.99$ to $1$.
• Figure 2: Characteristic function for $D=2$: (a) $m=0$ and (b) $m=1$. The squeezing parameter in both panels is $r=1$.
• Figure 3: Terms corresponding to the characteristic function of the squeezed multiplet for $D=3$, $m=0$, with $r=1.2$, shown in the panels discussed in the text. Panels (a), (b), and (c) show the direct (diagonal) terms, while panels (d), (e), and (f) correspond to the interference (non-diagonal) terms of the characteristic function. The full characteristic function, constructed as the sum of panels (a)–(f), is shown in panel (g).
• Figure 4: Examples characteristic functions of various multiplets with $r=1$. Columns correspond to $D=3$, $4$, $5$, and $6$, while rows correspond to $m=0$, $1$, and $2$.
• Figure 5: Upper-right quadrants of the characteristic functions for increasing $r=0.5$, $1$, $1.5$, and $2$, for fixed $D=4$ and $m=2$.