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Physical Review A

Excited-state quantum phase transitions in Kerr nonlinear oscillators

Qing-Wei Wang, Sha Wu

Published Dec 24, 2020• 21 citations
Publisher

Abstract

Excited-state quantum phase transitions (ESQPTs) generalize the concept of quantum phase transitions to excited levels. Here we investigate how such ESQPTs can emerge in Kerr nonlinear oscillators, where a bosonic mode with a Kerr nonlinearity is subject to both coherent (one-photon) and parametric (two-photon) driving. We determine the structure and properties of the complete phase diagram by defining proper dimensionless parameters. Rich phases and phase transitions exist in the phase diagram, including phases with two or four critical energies above the ground-state level and first-order quantum phase transitions. We characterize the ESQPTs by singularities in the density of states, which show universal logarithmic divergence or discontinuity at the critical energies. Furthermore, we find that observables such as the photon number also exhibit singular behaviors at the critical energies and obey scaling laws. These results may extend our understanding of the driven Kerr oscillator and ESQPTs in finite-component systems.

DOI
10.1103/physreva.102.063531

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Excited-state quantum phase transitions (ESQPTs) generalize the concept of quantum phase transitions to excited levels. Here we investigate how such ESQPTs can emerge in Kerr nonlinear oscillators, where a bosonic mode with a Kerr nonlinearity is subject to both coherent (one-photon) and parametric (two-photon) driving. We determine the structure and properties of the complete phase diagram by defining proper dimensionless parameters. Rich phases and phase transitions exist in the phase diagram, including phases with two or four critical energies above the ground-state level and first-order quantum phase transitions. We characterize the ESQPTs by singularities in the density of states, which show universal logarithmic divergence or discontinuity at the critical energies. Furthermore, we find that observables such as the photon number also exhibit singular behaviors at the critical energies and obey scaling laws. These results may extend our understanding of the driven Kerr oscillator and ESQPTs in finite-component systems.
We investigate and characterize the emergence of finite-component dissipative phase transitions (DPTs) in nonlinear photon resonators subject to n -photon driving and dissipation. Exploiting a semiclassical approach, we derive general results on the occurrence of second-order DPTs in this class of systems. We show that for all odd n , no second-order DPT can occur while, for even n , the competition between higher-order nonlinearities determines the nature of the criticality and allows for second-order DPTs to emerge only for n = 2 and n = 4 . As pivotal examples, we study the full quantum dynamics of three- and four-photon driven-dissipative Kerr resonators, confirming the prediction of the semiclassical analysis on the nature of the transitions. The stability of the vacuum and the typical timescales needed to access the different phases are also discussed. We also show a first-order DPT where multiple solutions emerge around zero, low, and high-photon numbers. Our results highlight the crucial role played by s t r o n g and w e a k symmetries in triggering critical behaviors, providing a Liouvillian framework to study the effects of high-order nonlinear processes in driven-dissipative systems, that can be applied to problems in quantum sensing and information processing.
We systematically investigate the excited-state quantum phase transition (ESQPT) in the anisotropic quantum Rabi model, which interpolates between the quantum Rabi model with Z2{\mathbb{Z}}_{2}Z2​ symmetry and the Jaynes-Cummings model with U(1)\mathbb{U}(1)U(1) symmetry. We calculate the model energy spectra and density of states (DOS) with the cumulants in both analytical and numerical ways to describe the ESQPT by the singularities. In the Jaynes-Cummings limit, its continuous U(1)\mathbb{U}(1)U(1) symmetry presents different nonanalytic behaviors from the quantum Rabi model with the discrete Z2{\mathbb{Z}}_{2}Z2​ symmetry, and there exists a finite discontinuous jump at the critical energy. For the general anisotropy case with Z2{\mathbb{Z}}_{2}Z2​ symmetry, there are two types of ESQPTs characterized by the finite discontinuous jump and the logarithmic divergence in the DOS, respectively. Different from the ground-state quantum phase transition, the ESQPT of the anisotropic quantum Rabi model strongly depends on the anisotropy which leads to a discontinuity even though the Z2{\mathbb{Z}}_{2}Z2​ symmetry is still preserved.
We propose concrete protocols to realize quantum criticality due to excited-state quantum phase transitions (ESQPTs) experimentally in presumably the simplest and most resilient system involving a single trapped ion oscillating in a radio-frequency Paul trap. We identify a specific class of excited states of the Extended Rabi Model (ERM) Hamiltonian, which occur between two critical ESQPT energies of the model in its (anti)Jaynes-Cummings superradiant phase. Properties of these states motivate the definition of several ESQPT witness observables. We study their critical scaling behaviors as well as various distinct state evolutions by driving the system across the quantum criticalities by changing the qubit-phonon coupling strength linearly in time at different finite rates. A mapping of the theoretical control parameters of the ERM to the experimental parameters of a trapped ion setup is provided, and simulations are performed for values referencing existing state-of-the-art setups, addressing both unitary state evolutions as well as relevant open-system corrections.
We study a minimal model that has a driven-dissipative quantum phase transition, namely, a Kerr nonlinear oscillator subject to driving and dissipation. Using mean-field theory, exact diagonalization, and the Keldysh formalism, we analyze the critical phenomena in this system, showing which aspects can be captured by each approach and how the approaches complement each other. Then critical scaling and finite-size scaling are calculated analytically using the quantum Langevin equation. The physics contained in this simple model is surprisingly rich: it includes a continuous phase transition, Z2{Z}_{2}Z2​ symmetry breaking, PT\mathcal{PT}PT symmetry, state squeezing, and critical fluctuations. Due to its simplicity and solvability, this model can serve as a paradigm for exploration of open quantum many-body physics.
The interplay between quantum criticality and ergodicity breaking constitutes a central challenge in complex quantum many-body systems. Here, we investigate ground-state quantum phase transitions (QPTs) and excited-state quantum phase transitions (ESQPTs), as well as ergodic-nonergodic transition in the cavity Heisenberg spin-chain (CHS) model. By combining quantum information measures with semiclassical fixed-point analysis, we identify a deformed phase that interpolates between the normal and superradiant phases, featuring a logarithmic nonanalyticity in the density of states (DoS) and two additional jump discontinuities. We further elucidate the spectral-structure mechanism underlying the connection between ESQPTs and the ergodic--nonergodic transition (ENET) via level statistics, participation ratios, and multifractal analysis. Our results provide a general framework for characterizing phase structures and spectral features in light--matter quantum many-body systems.