Nonlinear Phase Gates as Airy Transforms of the Wigner Function

TL;DR

This work addresses the challenge of implementing universal continuous-variable quantum computation with nonlinear phase gates by providing an analytic, phase-space description of cubic and quartic gates as Airy transforms of the Wigner function. The authors derive explicit Airy-transform expressions for unbounded gates and extend the framework to bounded, quartic scenarios (the quartic-bounded cubic gate), including multimode states, with careful treatment of scaling and nonlinear momentum maps. They show that the key non-Gaussian features, including Wigner negativity, persist under bounded gates and that the approach enables practical, analytic calculations for Gaussian inputs and their mixtures, while also analyzing the limitations and behavior in unbounded and inverted-potential regimes. The results establish a universal, analytically tractable toolkit for simulating and interpreting nonlinear quantum dynamics in phase space, with direct relevance to experimental realizations of cubic and quartic phase states and their use in continuous-variable quantum information processing.

Abstract

Low-order nonlinear phase gates allow the construction of versatile higher-order nonlinearities for bosonic systems and grant access to continuous variable quantum simulations of many unexplored aspects of nonlinear quantum dynamics. The resulting nonlinear transformations produce, even with small strength, multiple regions of negativity in the Wigner function and thus show an immediate departure from classical phase space. Towards the development of realistic, bounded versions of these gates we show that the action of a quartic-bounded cubic gate on an arbitrary multimode quantum state in phase space can be understood as an Airy transform of the Wigner function. This toolbox generalises the symplectic transformations associated with Gaussian operations and allows for the practical calculation, analysis and interpretation of explicit Wigner functions and the quantum non-Gaussian phenomena resulting from bounded nonlinear potentials.

Figures (7)

• Figure 1: Application of Airy transforms to analyse physical bounded nonlinear phase gates. The upper row shows the cubic phase gate $U_3$ with $\gamma_3=2$, and the quartic-bounded cubic phase gate $U_{3,4}$ with $\gamma_3=2$ and $\gamma_4=0.2$. The bottom row shows the TDW gate generated by the unitary operator $U_\text{TDW}=\exp[-\frac{i}{\hbar}(-18+15\hat{q}-\frac{7}{2}\hat{q}^2+\frac{0.2}{4}\hat{q}^4)]$ with the same $\gamma_4$, approximating the quartic bounded cubic potential. The effect of lower bounding the cubic gate is to limit the dynamics for negative position and momentum. The TDW is a poor substitute for the cubic phase gate at the level of phase space representation, also reflected in the nonlinear squeezing. All gates take the harmonic oscillator ground state as the initial state with $\hbar=1$ and the insets are the equivalent cubic, quartic and tilted double well potentials $V(q)$ forming the gates. For the nonlinear squeezing the black dashed line is the threshold for quantum non-Gaussianity, solid black is the harmonic oscillator ground state, blue (orange) is the (quartic-bounded) cubic phase state and green is the TDW state.
• Figure 2: Application of Airy transform to analyse nonlinearity accumulation with physical bounded gates. Action of the quartic bound cubic gate $(U_{3,4})^k$ (top) for $k=1,2,3$ left to right. Iteration of the gate increases the cubic effects, as seen by the re-emergence of the suppressed negative position and momentum region. This occurs even though at all stages the system is bounded from below by the quartic gate. The pure cubic phase states (bottom) have diverging momentum for both positive and negative momentum symmetrically. Initial states and parameters are as in Fig. \ref{['CvQ']}. The effective potentials corresponding to the applied gates are shown below the Wigner functions (left) where quartic-bounded cubic potentials are solid, cubic potentials are dashed. Increasing $k$ (blue, orange, green) leads to quartic-bounded cubic potentials that more closely approximate the cubic potential around the inflection point. However the increasing significance of the quartic term more strongly attenuates the nonlinear squeezing in absolute value in comparison with a cubic phase state (bottom right).
• Figure 3: Application of Airy transform to analyse a realistic cubic nonlinearity softening. The bare cubic gate compared with the cubic gate softened for $q>0$ by a weak inverted quartic nonlinearity. The diverging negative position and momentum due to the cubic gate are present in both examples. The effect of a weak inverted quartic gate is to suppress the positive position and negative momentum. The divergence is then faster in the region where both cubic and quartic potentials go to negative infinity together. Initial states and parameters are as in Fig. 1 of the main text.
• Figure 4: The Wigner functions for the cubic phase gate acting on various thermal states. Horizontally we set $\bar{n}=0, 0.5, 1$ and verically $\gamma_3=2$ (top), $\gamma_3=4$ (middle) and the quartic-bounded cubic gate (bottom) with $\gamma_3=2$ and $\gamma_4=0.2$. The nonclassical fluctuations are suppressed by the thermal noise, while the state spreads out in phase space, as seen by the extension of the parabolic 'arms'. Below these we show the Wigner cuts at $q=0$ with blue, yellow and green following the vertical ordering.
• Figure 5: The Wigner cut $W(0,p)$ for the cubic phase state with $\gamma=0.05,1$. The accuracy with respect to the analytical calculation is very good, although some values for $\gamma=0.05$ were erroneously reported as complex-valued (not shown). Curves from numerical integration are dashed. The lower values of $\gamma$ appear to require more time to estimate numerically.