Symmetries of the squeeze-driven Kerr oscillator

TL;DR

This work uncovers a hidden quasi-spin $su(2)$ symmetry in the squeeze-driven Kerr oscillator that emerges at integer values of the detuning-to-Kerr ratio $\eta = \Delta/K$, by analyzing the static effective Hamiltonian obtained from the boson expansion to second order. It reveals that the problem is governed by a spectrum-generating algebra $sp(2,\mathbb{R})$ (equivalently $su(1,1)$) with an alternative $u(2)$ description via an auxiliary boson, providing two complementary algebraic routes to diagonalization. The authors show that the $su(2)$ symmetry is remarkably robust to higher-order perturbations, especially the $P_2$ (pairing) term, while higher-order terms $P_3$ and $P_4$ introduce new degeneracy patterns and selection rules, with direct implications for stabilizing bosonic codes in quantum computing. They map a three-phase structure with associated separatrices and discuss excited-state quantum phase transitions (ESQPTs) in the spectrum, and outline extensions to multi-oscillator systems and to Limbladian dynamics, highlighting potential applications in robust quantum information processing with Kerr-based platforms.

Abstract

We study the symmetries of the static effective Hamiltonian of a driven superconducting nonlinear oscillator, the so-called squeeze-driven Kerr Hamiltonian, and discover a remarkable quasi-spin symmetry $su(2)$ at integer values of the ratio $η=Δ/K$ of the detuning parameter $Δ$ to the Kerr coefficient $K$. We investigate the stability of this newly discovered symmetry to high-order perturbations arising from the static effective expansion of the driven Hamiltonian. Our finding may find applications in the generation and stabilization of states useful for quantum computing. Finally, we discuss other Hamiltonians with similar properties and within reach of current technologies.

Figures (14)

• Figure 1: Excitation energy of the Hamiltonian (\ref{['Kerr_no_squeezing']}) as a function of the control parameter $\eta = \Delta/K$ (ratio of detuning to Kerr coefficient). The quasi-spin $| j,m \rangle$ classification of the degenerate states with energies $E(\eta =\textrm{odd})=m^{2}$, $E(\eta =\textrm{even})=m^{2}-\frac{1}{4}$ is included. Crossings are marked with blue dots and labelled by their associated $m$ values. The separatrix, marked with a dashed black line, is $E_{s}=\eta/2+\eta ^{2}/4$. The parity of the eigenstates is positive (orange) and negative (blue). The two phases are marked by blue and yellow backgrounds.
• Figure 2: Eigenvalues of the operator $-\hat{C}_{2}$ as a function of $v$ for $N=50$. Each point is doubly degenerate, $\pi^{\prime }=\pm$, except $v= N/2=25$.
• Figure 3: Eigenvalues of the operator $-\hat{P}_{2}$ as a function of $v$ for $N=50$.
• Figure 4: Excitation energy of Hamiltonian (\ref{['H_nodelta']}) as a function of the control parameter $\xi$. Quasi-spin $| j^{\prime },m^{\prime } \rangle$ classification of the degenerate states at the point of maximal rate of approach, $\xi_{c}=\pi v$ is shown. The separatrix, marked with a dashed black line, is $E_{s}=\xi^{2}$. The size of the Fock space is truncated at $N = 800$. States to the left of the separatrix are singly-degenerate with positive (blue) and negative (red) parity. States to the right of the separatrix are labeled by the number $v=0,1,2, \ldots$ of equation (\ref{['eq:v']}) and are doubly degenerate with parity $\pm.$
• Figure 5: Panel (a): The gap $(E_{v}^{(odd)}-E_{v}^{(even)})/K$ as a function of $\xi$ for $v=1,\ldots,12$. Panel (b): The derivative of the gap $-\frac{1}{K} \partial(E_{v}^{(odd)}-E_{v}^{(even)})/\partial \xi$ as a function of $\xi$ for $v=1,\ldots,12$. The critical value $\xi_{c}$ is the value of $\xi$ at the maximum of the derivative Frattini2022. The gaps go to zero as the separatrix is crossed for each $v$ as a function of $\xi$. To the right of the separatrix, states are doubly degenerate with $E_{v}^{(odd)}=E_{v}^{(even)}$ (parity $\pm$).