Spectral kissing and its dynamical consequences in the squeeze-driven Kerr oscillator

Abstract

Transmon qubits are the predominant element in circuit-based quantum information processing, such as existing quantum computers, due to their controllability and ease of engineering implementation. But more than qubits, transmons are multilevel nonlinear oscillators that can be used to investigate fundamental physics questions. Here, they are explored as simulators of excited state quantum phase transitions (ESQPTs), which are generalizations of quantum phase transitions to excited states. We show that the spectral kissing (coalescence of pairs of energy levels) experimentally observed in the effective Hamiltonian of a driven SNAIL-transmon is an ESQPT precursor. We explore the dynamical consequences of the ESQPT, which include the exponential growth of out-of-time-ordered correlators, followed by periodic revivals, and the slow evolution of the survival probability due to localization. These signatures of ESQPT are within reach for current superconducting circuits platforms and are of interest to experiments with cold atoms and ion traps.

Keywords: Quantum simulation; Qubits.

Fig. 1. Spectral kissing and localization.

a Energy levels as a function of the control parameter reproducing the experimental data with K/(2π) = 0.32 MHz and b $E^{\prime }/\left(\hslash K\right)$ for larger values of ξ. Solid lines are for the even parity sector and dashed lines for odd parity. The bright orange line in (b) marks the energy of the ESQPT, as given in Eq. (3). c–e Normalized density of states and f–h participation ratio for the eigenstates in the Fock basis for the values of ξ indicated in (c–e); even parity sector. Numerical (shade) and analytical (solid line) data are shown in (c–e). The vertical dashed line in (c–h) is the ESQPT energy from Eq. (3). i–l Husimi functions for different eigenstates and ξ = 180.

Fig. 2. Phase space and quantum dynamics.

a Energy curves in the phase space obtained with Eq. (2). The hyperbolic point is denoted as O, the center points are represented with blue diamonds, and the solid line intersecting at O is the separatrix. Points O, A–E mark the centers of the initial coherent states chosen for the quantum dynamics. b Evolution of the FOTOC, c Husimi entropy, and d survival probability as a function of time. The exponential [linear] curve with rate [slope] given by the Lyapunov exponent in Eq. (4) are indicated in (b)[c]. e Snapshots of the Husimi functions; each row refers to one of the six initial coherent states investigated, and each column to a different time, as indicated.