Time Crystals Reveal Abrupt Shifts Between States, Hinting at New Quantum Control Methods

Researchers have identified a dynamical phase transition within a collective-spin model displaying the unusual behaviour of a boundary time crystal. Sukrut Mondkar, Priya Ghosh, and Ujjwal Sen, all from the Harish-Chandra Research Institute, demonstrate this transition by driving the system across the boundary between the time crystal and non-time crystal phases using rapid or gradual changes to a key Hamiltonian parameter. Their analysis of the Loschmidt echo, a measure of initial state fidelity, reveals non-analytic features indicative of the transition, and importantly, establishes the persistence of this dynamical quantum phase transition even with slower, more controlled driving protocols. This work is significant because it clarifies the robustness of time crystal phases and provides insights into non-equilibrium dynamics in driven quantum systems, potentially informing future developments in quantum technologies.

This research demonstrates the existence of a DQPT by initializing the system in either the BTC or a non-BTC phase and subsequently driving it across the transition point.

The driving is achieved through either an abrupt quench or a finite-time linear ramp of a Hamiltonian control parameter, governed by Markovian Lindblad dynamics. Diagnosing these DQPTs involves analysing zeros in the fidelity-based Loschmidt echo, a measure of the overlap between the initial state and the evolving mixed state, which manifest as non-analytic cusp-like features in the associated rate function.
For quenches leading into the BTC phase, the Loschmidt echo displays a series of zeros directly linked to the emergent time-periodic steady state of the system. Conversely, quenches into the non-BTC phase result in a complete loss of overlap, with the system remaining in a stationary state after relaxation.

Further investigation confirms that this DQPT persists even when employing a ramp protocol followed by unitary evolution with the final Hamiltonian. The study extends to an analysis of finite-size scaling, specifically examining the first critical time. Results indicate convergence towards a constant value in the thermodynamic limit, with distinct power-law behaviours observed for both the quench and ramp protocols.

This work establishes a connection between dynamical quantum phase transitions and the unique time-crystalline order found in open quantum systems, opening avenues for exploring their interplay and potential applications. The findings reveal qualitative differences in the late-time dynamics, providing insights into the behaviour of these non-equilibrium phases of matter.

Loschmidt echo analysis of dynamical phase transitions in a driven boundary time crystal reveals fragile many-body localization

Researchers investigated dynamical quantum phase transitions (DQPTs) within a dissipative collective-spin model exhibiting the boundary time crystal (BTC) phase. The study initialized the system in either the BTC or non-BTC phase ground state, subsequently driving it across the BTC transition using two distinct methods.

These methods involved either an abrupt quench or a finite-time linear ramp of a Hamiltonian control parameter, both governed by Markovian Lindblad dynamics. Diagnosis of DQPTs relied on analysing zeros of the fidelity-based Loschmidt echo, calculated between the initial state and the evolving mixed state.

These zeros manifest as nonanalytic cusp-like features within the associated rate function, serving as key indicators of the phase transition. For quenches driving the system into the BTC phase, the Loschmidt echo displayed zeros attributable to the emergence of a time-periodic steady state. Conversely, quenches into the non-BTC phase resulted in a vanishing overlap that remained zero as the dynamics relaxed to a stationary state.

Further analysis demonstrated the persistence of the DQPT even when employing the ramp protocol, followed by unitary evolution with the final Hamiltonian. The finite-size scaling of the first critical time was then examined, revealing convergence to a constant value in the thermodynamic limit. Distinct power-law approaches were observed for the quench and ramp protocols, highlighting the influence of the driving method on the system’s behaviour. This meticulous analysis of both quench and ramp dynamics provides a comprehensive understanding of DQPTs in boundary time crystals.

Dynamical Phase Transitions Identified via Loschmidt Echo and Rate Function Analysis reveal critical slowing down

Zeros of the fidelity-based Loschmidt echo were observed, indicating a dynamical phase transition in a dissipative collective-spin model exhibiting a boundary time crystal (BTC) phase. The research details the emergence of this transition through abrupt quenches and finite-time linear ramps of a Hamiltonian control parameter under Markovian Lindblad dynamics.

Diagnostic tools revealed non-analytic cusp-like features in the associated rate function, pinpointing the DQPTs. For quenches driving the system into the BTC phase, the Loschmidt echo exhibited repeated zeros at a sequence of critical times, directly linked to the emergent time-periodic steady state.

Conversely, quenches into the non-BTC phase resulted in the overlap vanishing and remaining zero after the dynamics relaxed to a stationary state. This qualitative distinction in late-time dynamics serves as a key signature of the DQPT. Further analysis confirmed the persistence of the DQPT under ramp protocols, followed by unitary evolution with the final Hamiltonian.

Finite-size scaling analysis of the first critical time demonstrated convergence to a constant value in the thermodynamic limit. Distinct power-law approaches were identified for both the quench and ramp protocols, providing insight into the scaling behaviour of the transition. The study employed a collective-spin model with N spin-1/2 particles, defined by a Hamiltonian and Lindblad operator, with all physical quantities expressed in units of K = 1. Parameters ω0, ωx, ωz, and κ were utilized to define the BTC and non-BTC phases, with the average magnetization ⟨Sz⟩/N oscillating in the BTC phase and approaching a stationary value in the non-BTC phase.

Loschmidt echo analysis confirms dynamical phase transitions and time-crystal behaviour in driven quantum systems

Scientists have demonstrated dynamical quantum phase transitions within a dissipative collective-spin model exhibiting the boundary time crystal phase. Investigations involved initializing the system in either the boundary time crystal or a non-boundary time crystal phase, then driving it across the transition using both abrupt quenches and finite-time linear ramps under Markovian Lindblad dynamics.

The presence of dynamical quantum phase transitions was diagnosed through analysis of the fidelity-based Loschmidt echo, revealing non-analytic cusp-like features in the associated rate function. Specifically, zeros in the Loschmidt echo indicated the emergence of a time-periodic steady state following quenches into the boundary time crystal phase, while overlap vanishing and remaining zero signified relaxation to a stationary state in the non-boundary time crystal phase.

Finite-size scaling analysis of the first critical time converged to a constant value in the thermodynamic limit, exhibiting distinct power-law behaviour for the quench and ramp protocols. The research extends the understanding of dynamical quantum phase transitions to genuinely dissipative time-crystalline phases, moving beyond closed or Floquet systems.

The authors acknowledge limitations stemming from the restricted dataset size, which limited the ability to fit certain parameters with high precision. To address this, they employed a manual parameter variation technique to minimize fitting errors. Future research may focus on exploring these phenomena with larger system sizes and more complex driving protocols to further refine the understanding of dynamical quantum phase transitions in dissipative systems and boundary time crystals. These findings establish a foundation for investigating non-equilibrium dynamics and phase transitions in open quantum many-body systems.