[Paper Review] Observation of first- and second-order dissipative phase transitions in a two-photon driven Kerr resonator

Guillaume Beaulieu, Fabrizio Minganti+5 more|arXiv (Cornell University)|Oct 20, 2023

Quantum Information and Cryptography4 citations

TL;DR

This study presents the first experimental observation of both first- and second-order dissipative phase transitions (DPTs) in a two-photon driven superconducting Kerr resonator. By engineering a parametrically driven nonlinear cavity, the authors observe phase coexistence and hysteresis in first-order DPTs, along with spontaneous symmetry breaking and critical slowing down in second-order DPTs, validated through quantum trajectory monitoring and Liouvillian spectral analysis, demonstrating control over non-equilibrium criticality in superconducting circuits.

ABSTRACT

In open quantum systems, first- and second-order dissipative phase transitions (DPTs) can emerge in the thermodynamic limit from the competition between unitary evolution, driving terms, and dissipation. The order of a DPT is defined by the continuity properties of the steady state. Until now, second-order DPTs have predominantly been investigated theoretically, while first-order DPTs have been observed in key experiments based on the theory of the single-photon driven Kerr resonator. We present here the first comprehensive experimental and theoretical analysis of both first and second-order DPTs in a two-photon (i.e., parametrically) driven Kerr superconducting resonator. Firstly, we characterize the steady state and its main features at the second- and first-order critical points: squeezing below vacuum and coexistence of two phases with different photon numbers, respectively. Then, by continuously monitoring the system along quantum trajectories, we study the non-equilibrium dynamics across the critical points. We witness the hysteresis cycles associated with the first-order DPT and the spontaneous symmetry breaking due to the second-order DPT. Applying the spectral theory of the Liouvillian superoperator, we develop efficient procedures to quantify the critical slowing down associated with the timescales of these processes. When scaling towards the thermodynamic limit, these timescales span five orders of magnitude. Our results corroborate the predictions derived using the Liouvillian theory of DPTs. This work stands as a compelling example of engineering and controlling of criticality in superconducting circuits. It marks a significant advancement in the use of two-photon driven Kerr resonators for criticality-enhanced quantum information applications.

Motivation & Objective

• To experimentally observe and characterize both first- and second-order dissipative phase transitions (DPTs) in a two-photon driven superconducting Kerr resonator.
• To investigate the non-equilibrium dynamics across critical points using continuous quantum trajectory monitoring.
• To quantify critical slowing down via the spectral theory of the Liouvillian superoperator.
• To validate theoretical predictions of DPTs in open quantum systems using a scalable superconducting circuit platform.
• To demonstrate engineering of criticality for potential applications in quantum information and enhanced sensing.

Proposed method

• Implementation of a $λ/4$ coplanar waveguide resonator capacitively coupled to a feedline for signal collection and terminated via a SQUID to tune resonance frequency and Kerr nonlinearity.
• Application of a coherent pump tone at approximately twice the cavity frequency to induce a two-photon drive, enabling parametric excitation.
• Use of heterodyne detection to continuously monitor the system along quantum trajectories, enabling observation of non-equilibrium dynamics.
• Spectral analysis of the Liouvillian superoperator to extract the smallest non-zero eigenvalues, quantifying critical slowing down timescales.
• Numerical and analytical modeling of the Lindblad master equation to simulate steady-state properties and correlation functions.
• Comparison of measured quantum trajectories with theoretical predictions to extract the Liouvillian gap and confirm critical behavior.

Experimental results

Research questions

• RQ1 Can first- and second-order dissipative phase transitions be experimentally observed in a two-photon driven Kerr resonator?
• RQ2 How do the non-equilibrium dynamics, including hysteresis and spontaneous symmetry breaking, manifest across critical points?
• RQ3 What is the role of the Liouvillian spectrum in characterizing critical slowing down in open quantum systems?
• RQ4 How do the timescales of relaxation and decoherence scale toward the thermodynamic limit in such systems?
• RQ5 To what extent can quantum trajectory monitoring reveal the underlying critical behavior of dissipative phase transitions?

Key findings

• The experiment observes a first-order DPT characterized by phase coexistence, metastability, and hysteresis, confirmed by quantum trajectory monitoring.
• A second-order DPT is observed with spontaneous symmetry breaking, evidenced by the emergence of a non-zero order parameter and critical fluctuations.
• The Liouvillian spectral gap associated with the second-order DPT (λ_SSB) and the first-order DPT (λ_1st) are experimentally extracted via quantum trajectory analysis, confirming theoretical predictions.
• Critical slowing down is quantified through the exponential decay of correlation functions, with timescales spanning five orders of magnitude when scaling toward the thermodynamic limit.
• The system exhibits squeezing below vacuum in the steady state at the second-order critical point, indicating strong quantum correlations.
• The agreement between measured quantum trajectories and Liouvillian spectral theory validates the theoretical framework for DPTs in open quantum systems.