Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Summary

Table of Links

• Introduction
• Model and Protocol
• Different Regimes in d˜− Γ˜ Plain and Multiple QMPE
• QMPE in Region (d): Second-order Ep
• QMPE in Region (a1): Oscillations
• Summary
• Appendix A: Eigenvectors of L in region (d)
• Appendix B: Eigenvectors of L in region (a1)
• Appendix C: QMPE in region (b): purely exponential relaxation
• Appendix D: QMPE in region (e): 3rd order EP
• References

VI. SUMMARY

We have studied QMPE in a two-level quantum system subjected to an oscillatory electric field and dissipative coupling with the environment. The Lindbladian of this two-level driven dissipative system, termed as Hatano’s model, has rich variety of eigen-spectrum, especially EPs and complex eigenvalues which are generic features of non-Hermitian quantum systems. We have investigated the roles of complex eigenvalues and EPs on the QMPE. The exact analytical formulae obtained for time evolved density matrix elements enable us to propose analytical criteria for the QMPE in observables. Interestingly, at the EPs, the density matrix elements and consequently observables (e.g. energy) attain additional algebraic time dependent relaxations apart from the usual exponential relaxations. Such additional algebraic time dependence at the EPs leads to double QMPE (twice intersections between trajectories starting at different initial conditions and relaxing towards the same steady state). We have procured the exact expressions for intersection times of trajectories, that mark the occurrence of QMPE and we provide analytical criteria that demarcate control parameters’ regimes with single or double or no QMPE. Whereas the multiple QMPE at EPs in the present model is restricted to maximum double QMPE i.e. two intersections, the multiple QMPE at the complex eigenvalues can result in even more number of intersections in a oscillatory manner. Notably, we have established the existence of thermal QMPE (single QMPE at EPs and multiple QMPE at complex eigenvalues), in presence of coherence in Hatano’s model. It is fascinating that, similar to energy and entropy, temperature also exhibits multiple QMPE where the initially hotter and colder copies intersect each other multiple times. Such multiple reversals of identities (i.e. hotter becoming colder and vice versa) result in multiple thermal QMPE. In spite of achieving multiple QMPE in several observables (ground state probability, energy, entropy, temperature) using either additional algebraic time dependence at EPs or oscillations at complex eigenvalues, surprisingly the KL divergence (a distance measure from steady state) shows single QMPE only. Overall, our theoretical analysis exhibits that EPs and complex eigenvalues can be used as systematic routes to generate multiple QMPE. However, we should mention that the presence of EPs or oscillations are not necessary for multiple QMPE. In this connection, we have shown that multiple QMPE can be achieved even with purely exponential relaxations, by putting suitable constraints on the gap statistics of the eigen-spectrum (Appendix C).

An important and generic future goal in QMPE research would be to understand its utility in speeding up quantum processes. In fact, a recent work on accelerating relaxation in open quantum systems by engineering EPs [84] hints towards the connection of such acceleration to QMPE. Our present work analyzes in detail how EPs lead to QMPE and provide criteria on control parameters to observe the QMPE. It is important to find out the deeper connections between the QMPE and faster relaxations at EPs and for generic open quantum systems. In this connection, an intriguing question is to find how the quantum speed limit may affect the degree of faster relaxation that can be achieved through the QMPE. Our work on QMPE by controlling the amplitudes of oscillatory electric fields hints towards possible experimental realizations of the QMPE in qubit, keeping in mind the recent advancement in qubit related experiment [119]. The analysis of QMPE in the present study paves path for future investigations of QMPE in quantum many-body systems.

Acknowledgements.- We thank Naomichi Hatano, Sosuke Ito, Ryo Hanai, Arnab Pal, Naruo Ohga and Kohei Yoshimura for useful discussions. This work is partially supported by the Grants-in-Aid for Scientific Research (Grant No. 21H01006 and No. 20K14428). A.K.C. gratefully acknowledges postdoctoral fellowship from the YITP. The numerical calculations have been done on Yukawa-21 at the YITP.