[Paper Review] Indication of critical scaling in time during the relaxation of an open quantum system

Ling-Na Wu, Jens Nettersheim+7 more|arXiv (Cornell University)|Aug 10, 2022

Opinion Dynamics and Social Influence4 citations

TL;DR

This paper demonstrates critical scaling in time during the relaxation of an open quantum system, where the entropy of a Caesium atomic spin state transiently peaks near maximum value due to dissipative coupling with a rubidium Bose gas. Finite-size scaling analysis reveals a critical point at a specific time, characterized by universal critical exponents independent of system details, indicating dynamical criticality in open, non-isolated quantum systems.

ABSTRACT

Phase transitions correspond to the singular behavior of physical systems in response to continuous control parameters like temperature or external fields. Near continuous phase transitions, associated with the divergence of a correlation length, universal power-law scaling behavior with critical exponents independent of microscopic system details is found. Recently, dynamical quantum phase transitions and universal scaling have been predicted and also observed in the non-equilibrium dynamics of isolated quantum systems after a quench, with time playing the role of the control parameter. However, signatures of such critical phenomena in time in open systems, whose dynamics is driven by the dissipative contact to an environment, were so far elusive. Here, we present results indicating that critical scaling with respect to time can also occur during the relaxation dynamics of an open quantum system described by mixed states. We experimentally measure the relaxation dynamics of the large atomic spin of individual Caesium atoms induced by the dissipative coupling via spin-exchange processes to an ultracold Bose gas of Rubidium atoms. For initial states far from equilibrium, the entropy of the spin state is found to peak in time, transiently approaching its maximum possible value, before eventually relaxing to its lower equilibrium value. Moreover, a finite-size scaling analysis based on numerical simulations shows that it corresponds to a critical point with respect to time of the dissipative system in the limit of large system sizes. It is signalled by the divergence of a characteristic length at a critical time, characterized by critical exponents that are found to be independent of system details.

Motivation & Objective

• To investigate whether critical phenomena in time—previously observed only in isolated quantum systems—can emerge in open quantum systems with dissipative dynamics.
• To explore if transient entropy peaks in non-equilibrium relaxation of open systems signal a critical point in time, analogous to equilibrium phase transitions.
• To determine whether universal critical exponents emerge in such open systems, independent of microscopic details.
• To establish a connection between time-resolved entropy dynamics and critical scaling, using finite-size scaling analysis.

Proposed method

• Experimental realization of a spin-3 system in individual Caesium atoms coupled to a spin-polarized rubidium Bose gas via inelastic spin-exchange collisions.
• Time-resolved measurement of entropy evolution in individual atoms following non-equilibrium initial states, observing transient entropy peaks near maximum possible value.
• Numerical simulations of the master equation for spin dynamics under dissipative coupling, modeling both unidirectional and bidirectional spin transitions.
• Finite-size scaling analysis of entropy peak time and localization length across system sizes, identifying a critical point at a specific time.
• Derivation of effective hydrodynamic equations for probability distribution and entropy using time-dependent perturbation theory in the continuous limit.
• Use of inverse temperature β and localization length ξ to probe critical behavior, showing scaling with βΔM and universal functional dependence.

Experimental results

Research questions

• RQ1 Can critical scaling in time emerge in open quantum systems undergoing relaxation, despite dissipative coupling to an environment?
• RQ2 Does the transient entropy peak in the relaxation dynamics correspond to a critical point in time, analogous to equilibrium phase transitions?
• RQ3 Are the critical exponents observed in the time-resolved dynamics universal, independent of system-specific parameters such as coupling strength or initial state?
• RQ4 How does the localization length ξ scale with system size and time near the entropy peak, indicating criticality?
• RQ5 Can the effective thermodynamic quantities like inverse temperature β and specific heat be expressed as universal functions of βΔM near the critical time?

Key findings

• The entropy of the Caesium spin state transiently reaches S_peak ≈ 1.944 ≈ ln(7) = S_max, indicating a peak in mixedness near maximum possible entropy.
• Finite-size scaling analysis shows that the entropy peak occurs at a critical time t_c, with the localization length ξ diverging as system size M increases.
• Critical exponents characterizing the divergence of ξ and the scaling of entropy and energy are found to be universal, independent of system details such as initial state or coupling strength.
• The effective inverse temperature β and localization length ξ scale universally with βΔM, indicating a universal critical behavior near the peak time.
• Numerical simulations confirm that the system exhibits critical scaling with time as the control parameter, with the effective dynamics governed by a universal function of βΔM.
• The entropy deviation from maximum, 1 - S/S_max, scales quadratically with (βΔM)^2, consistent with Gibbs state assumptions and analytical predictions.