Constructing the generalized Gibbs ensemble after a quantum quench (1203.0901v5)

Abstract: Using a numerical renormalization group based on exploiting an underlying exactly solvable non- relativistic theory, we study the out-of-equilibrium dynamics of a 1D Bose gas (as described by the Lieb-Liniger model) released from a parabolic trap. Our method allows us to track the post-quench dynamics of the gas all the way to infinite time. We also exhibit a general construction, applicable to all integrable models, of the thermodynamic ensemble that has been suggested to govern this dynamics, the generalized Gibbs ensemble. We compare the predictions of equilibration from this ensemble against the long time dynamics observed using our method.

• The paper introduces a novel numerical renormalization group method to construct and validate the generalized Gibbs ensemble in integrable quantum systems.
• It demonstrates that incorporating non-trivial conserved quantities into the GGE yields more accurate predictions for long-term dynamics than traditional ensembles.
• The study offers practical insights for ultra-cold atom experiments by highlighting the role of integrability in the non-equilibrium behavior of quantum gases.

Overview of the Generalized Gibbs Ensemble after Quantum Quench

The paper by Jean-Sebastien Caux and Robert M. Konik presents a detailed paper on the non-equilibrium dynamics of a one-dimensional Bose gas using the Lieb-Liniger model released from a parabolic trap. The focus is on constructing and validating the generalized Gibbs ensemble (GGE) as a thermodynamic ensemble to describe the equilibration process post-quench. Through employing sophisticated numerical renormalization group techniques tailored to exploit integrable models, the authors offer comprehensive insights into both the initialization and long-term dynamics of the system, paving the way for a deeper understanding of quantum systems far from equilibrium.

Key Findings

• Numerical Approach: A numerical renormalization group based on an exactly solvable nonrelativistic framework successfully tracks the gas dynamics over time, including the infinite-time limit. The approach goes beyond standard methodologies like time-dependent density matrix renormalization group (tDMRG) by leveraging integrability.
• Generalized Gibbs Ensemble: The authors propose an innovative construction of the GGE that accounts for non-trivial conserved quantities, enhancing the traditional Gibbs ensemble's descriptive power concerning integrable systems' dynamics. The ensemble incorporates a complete set of conserved quantities $Q_i$ and generalized temperatures $\beta_i$:

$$\hat{\rho}_{GGE} = Z^{-1} \exp \left(-\sum_i \beta_i Q_i\right)$$

• Equilibration Analysis: The GGE's predictions are juxtaposed against long-term dynamics observed numerically. For lower momenta, the ensemble offers a more accurate portrayal than the traditional grand canonical ensemble (GCE), albeit some deviations remain at finite momentum. This contrasting behavior of ensembles at different momenta suggests potentially missing correlations associated with non-local properties in the GGE.

Practical and Theoretical Implications

• Integrable Systems: This research underscores the significance of integrable models in describing quantum systems under non-equilibrium conditions. The presented framework could readily extend to other models like the Heisenberg and XXZ spin chains, further simplifying complex dynamical analyses.
• Quantum Gases and Ultra-cold Atoms: The implications for ultra-cold atomic experiments are noteworthy. Understanding non-equilibrium dynamics in terms of the GGE can aid in designing better control protocols for quantum simulations, informing experimental settings where traditional thermalization is not observed.

Speculation on Future Developments

The paper opens avenues for exploring the exact mechanisms by which integrable systems retain memory of their initial states, potentially impacting broader quantum computing applications. Future studies might focus on further elucidating how non-local correlations alter the GGE's predictive accuracy, providing insights into integrating quantum systems and the computational algorithms that simulate their behaviors.

In summary, the paper delivers a rigorous examination of out-of-equilibrium dynamics, articulating a refined approach to characterizing these systems through the generalized Gibbs ensemble. It establishes itself as a critical resource for researchers pursuing the intricate mechanics of quantum systems beyond traditional equilibrium frameworks.