Stabilization of cat-state manifolds using nonlinear reservoir engineering

TL;DR

This work introduces Nonlinear Reservoir Engineering (NLRE), a design framework that stabilizes multi-component Schrödinger cat-state manifolds by engineering destructive interference between nonlinear gain and loss processes. The central construct is the jump operator ${\hat{K}=\hat{a}^{\dagger\,r} f(\hat{n})-g(\hat{n})\hat{a}^{l}}$, whose crossing point ${k^*}$ with height ${h^*}$ determines a $d=r+l$-dimensional dark-state manifold with discrete rotational symmetry, enabling a rich class of rotation-symmetric bosonic codes. The paper analyzes steady-state distributions, error-correction capabilities (including autonomous QEC and passive symmetry protection), and provides concrete implementations in trapped ions and circuit QED (with nonlinear elements such as an ATS) while connecting to nonlinear coherent states and generalized Bogoliubov mappings. It further shows how NLRE can realize rotation-symmetric codes beyond standard cat codes, including quadrature-squeezed cat manifolds, and develops a generalization framework via unitary transformations to map stabilization schemes to different error channels. The results offer a practical, non-perturbative route to engineer robust bosonic codes across platforms and suggest broader applicability to non-Gaussian quantum states and error-correcting architectures.

Abstract

We introduce a novel reservoir engineering approach for stabilizing multi-component Schrödinger's cat manifolds. The fundamental principle of the method lies in the destructive interference at crossings of gain and loss Hamiltonian terms in the coupling of an oscillator to a zero-temperature auxiliary system, which are nonlinear with respect to the oscillator's energy. The nature of these gain and loss terms is found to determine the rotational symmetry, energy distributions, and degeneracy of the resulting stabilized manifolds. Considering these systems as bosonic error-correction codes, we analyze their properties with respect to a variety of errors, including both autonomous and passive error correction, where we find that our formalism gives straightforward insights into the nature of the correction. We give example implementations using the anharmonic laser-ion coupling of a trapped ion outside the Lamb-Dicke regime as well as nonlinear superconducting circuits. Beyond the dissipative stabilization of standard cat manifolds and novel rotation symmetric codes, we demonstrate that our formalism allows for the stabilization of bosonic codes linked to cat states through unitary transformations, such as quadrature-squeezed cats. Our work establishes a design approach for creating and utilizing codes using nonlinearity, providing access to novel quantum states and processes across a range of physical systems.

Figures (16)

• Figure 1: Toy model of the nonlinear reservoir engineering method. (a) The interaction in Eq. \ref{['eq:hamilton_modif']} is engineered such that the bosonic mode is simultaneously subject to competing nonlinear boson raising $\tilde{f}$ and lowering $\tilde{g}$ processes of orders $r$ and $l$, respectively. (b) The boson distribution is stabilized around the crossing of the strengths (termed Rabi frequency) of $\tilde{f}$ and $\tilde{g}$. (c) The dissipative dynamics stabilizes a manifold of dimension ${d=r+l}$ around the crossing point (here, $r=1$ and $l=2$) with basis states $\ket{\Xi_\mu}$ that have Fock state population occupied every $d$ states. (d) Classical phase-space trajectories. The stable critical points of the classical dynamics are highlighted by red dots, suggesting the existence of $d$ attractors exhibiting rotational symmetry relative to the origin. (e) Wigner quasiprobability of steady states for the processes $\tilde{f}$ and $\tilde{g}$ shown in (c) for various orders. The dimension $d$ sets the number of nonlinear coherent states composing the stabilized state.
• Figure 2: Change from five to two steady states as the state's variance increases. The lower panel illustrates the real part of the nine lowest eigenvalues $\Lambda_i$ of the effective Liouvillian $\mathcal{L}$ for the case ${(r,l)=(3,2)}$ with $\tilde{f}$ and $\tilde{g}$ as in Fig. \ref{['fig1:rabi_frequency']}. The system exhibit two dark states associated to $\Lambda_{1/2}=0$ and three metastable states associated to $\Lambda_{3/4/5}\approx0$. Leakage from the latter into the former starts when the steady state predicted solely by Eq. \ref{['eq:recusive_relation']} has some non negligible population in the first five Fock states. The lower broken axis illustrates the real parts of the eigenvalues $\Lambda_{6/\ldots/9}$ corresponding to the first excited eigenstates of the Liouvillian.
• Figure 3: Shape of the stabilized states. (a) Illustration of the parameters affecting the form of the steady states, specifically the angle between the functions $\tilde{f}$ and $\tilde{g}$, and the value $h^*$ of the Rabi frequency at the intersection point $k^*$. (b) Classification of the shapes into three primary categories based on the Mandel $Q$ parameter: number-squeezed states for $Q<0$, coherent shaped states for $Q=1$, and phase-squeezed shaped states for $Q>0$. (c) The shape can be modified by varying the angle $\vartheta_f+\vartheta_g$ between the functions. An increase in the ratio of $\vartheta_f$ to $\vartheta_g$ impacts the skewness of the stabilized boson distribution and decreases $Q$. The shape of the steady state is also governed by $h^*$, with larger values resulting in an increase in $Q$. Theoretical curves were obtained using variance and mean boson number given in Eq. \ref{['eq:mean_and_variance_CMB']}. In all situations $k^*=10$ and $(r,l)=(0,2)$, on the left $h^*=20$ while on the right $\vartheta_f=3\vartheta_g$.
• Figure 4: (a) Standard dissipative cat-state stabilization explained using the nonlinear reservoir engineering method. Rabi frequencies and stabilized boson distribution of the jump operator ${\hat{a}^3-\alpha^3}$ with $\alpha=3.5$. (b) Relative entropy of a Poisson distribution, parameterized by $\alpha$, with respect to its approximation using a CMP distribution given in Eq. \ref{['eq:CMP_distribution']}. The Poisson distribution is stabilized using ${\hat{a}^d-\alpha^d}$, where $d$ represents the dimension of the coherent-state manifold. The low relative entropy suggests that the true and approximate distributions contain nearly identical amounts of information. This demonstrates that the proposed nonlinear reservoir engineering method effectively generalizes well-established reservoir engineering cases.
• Figure 5: (a) We evaluate the effective confinement rate $\kappa_\mathrm{conf}$ as the exponential rate at which a dark state initially displaced ($\delta x = 0.001$) returns to the stabilized manifold. (b) Effective confinement rate with respect to the angle $\vartheta_f+\vartheta_g$ between the functions $\tilde{f}$ and $\tilde{g}$. One can increase this rate by increasing the ratio $\vartheta_f/\vartheta_g$ which effectively increases the skewness of the stabilized boson distribution (the height is constant $h^*=20$). The theoretical curves have been obtained using Eq. \ref{['eq:confine_rate']} additionally corrected for the skewness. (c) The relationship between the height of the crossing point $h^*$ and $\kappa_\mathrm{conf}$ is linear (${\vartheta_f=\vartheta_g=\pi/3}$). In both (b) and (c), we keep the crossing point at $k^*=20$.