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Nonequilibrium steady states in the Floquet-Lindblad systems: van Vleck's high-frequency expansion approach

by Tatsuhiko N. Ikeda, Koki Chinzei, Masahiro Sato

Submission summary

As Contributors: Tatsuhiko Ikeda
Arxiv Link: (pdf)
Date submitted: 2021-07-26 02:37
Submitted by: Ikeda, Tatsuhiko
Submitted to: SciPost Physics Core
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


Nonequilibrium steady states (NESSs) in periodically driven dissipative quantum systems are vital in Floquet engineering. Here, for high-frequency drives with Lindblad-type dissipation, we develop a general theory to characterize and analyze NESSs based on the high-frequency (HF) expansion without numerically solving the time evolution. This theory shows that NESSs can deviate from the Floquet-Gibbs state depending on the dissipation type. We show the validity and usefulness of the HF-expansion approach in concrete models for a diamond nitrogen-vacancy (NV) center, a kicked open XY spin chain with topological phase transition under boundary dissipation, and the Heisenberg spin chain in a circularly-polarized magnetic field under bulk dissipation. In particular, for the isotropic Heisenberg chain, we propose the dissipation-assisted terahertz (THz) inverse Faraday effect in quantum magnets. Our theoretical framework applies to various time-periodic Lindblad equations that are currently under active research.

Current status:
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Submission & Refereeing History

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Submission 2107.07911v2 on 26 July 2021

Reports on this Submission

Anonymous Report 1 on 2021-9-3 (Invited Report)


1 - The Floquet-Lindblad equation analysed in this work is important for many systems of current interest. The inclusion of dissipation is relevant for open quantum systems and solid state systems.

2 - The expansion allows to calculate the NESS at high frequency just by looking at the spectrum of the effective Liouvillian $\mathcal{L}_{\textrm{eff}}$, which can be systematically computed by way of the expansion. No study of the transient to the NESS is required.

3 - The properties of the expansion and of the NESS are clearly presented. The effectiveness of the method is shown for several model.

4 - The state of the art in the field is properly addressed and the relevant literature is well referenced.

5 - In the case of time dependent dissipators, it is shown that the NESS coincide with the "quasi-thermal" Floquet-Gibbs state when the spectral width of the bath is much smaller than the drive frequency. This can have important applications to electronic systems driven with high frequency lasers.


1 - The high $\omega$ expansion does not seem to be easily tractable analytically, in general. The authors do not present a fully analytic study of a sample system: for the simplest example (XY chain) the effects of dissipation on the phase diagram are treated qualitatively.

2 - It is not clear wether a general rule exists to quantify the error of the expansion, similarly to Eq. (66).

3 - The conclusions present a limited outlook view.

4 - Figure 3, 4 and 6 are not very legible.


In this paper the authors present a systematic high-frequency expansion of the quantum master equation to study the non-equilibrium steady state of a Floquet-Lindblad system subject to periodic drives and to dissipation, and apply it to selected models.

The topic is of current relevance, given the advances in the study of driven systems, as properly shown in the cited literature. The study of dissipation is also important for many solid-state systems where the driven degrees of freedom are coupled to dissipative baths of various kinds.

The authors review the most common techniques used to study a NESS, and nicely present the generalization of the Van Vleck expansion (Ref. 8 and 9) to dissipative Lindblad equations. The properties of the equations and of the NESS are clearly stated.
The authors then apply their findings to an XY chain, a three level system modelling NV centres, and to the inverse Faraday effect, study the properties of the NESS and the applicability of the expansion.

The article is mostly well written and orderly presented, especially in Sec. 1-4. The properties of the expansion are nicely stated in section 5.1 and 6.1, but the presentation of the results for model systems is less organized. The appendices are very useful in understanding the main text.

I have a few remarks.

a) The high-expansion analysis of Sec. 5.2 is performed on the hamiltonian evolving $W_k$ (Majorana fermions), and it is not explained how to relate it to a high-frequency expansion for the spin density matrix.

b) The phase diagram in Fig. 1 changes drastically from 0th to 2nd order expansion. Is an expansion to 2nd order sufficient to get the full picture of the physics involved?

c) It is not immediately clear to me why Eq. (50) implies property (ii).

d) In Eqs. (51)-(55), the dissipator $\mathcal D$ commutes with $\mathcal S_z$ independently of the sites on which the jump operators act. Therefore, while it is true that the drive prevents an edge dissipator from propagating into the bulk, Eq. (50) would still hold even for a dissipator acting on the entire chain. Would property (ii) still be true in that case or would the phase diagram change?

e) Would $\gamma_{\alpha\beta}(\epsilon_m-\epsilon_n-k\omega)$ in Eq. (82) have a simple representation in terms of Feynmann diagrams of the dissipative process?

Overall the work is good and the novelty and usefulness of the method presented are sufficiently demonstrated to warrant publication in SciPost Physics Core. Thus I recommend publication of the manuscript once the issues raised in this report are properly addressed.

Requested changes

1 - Add a (brief) discussion of what would change in the results of Sec. 5.2 if the dissipation did not occur on the edges (remark d).

2 - Add an explanation of why exactly Eq. (50) implies property (ii) (remark c).

3 - Add a better connection between Appendix C and Eq. (49) (remark a).

4 - Fix the panel labels in Figure 10, 11 and 12.

5 - Improve the readability of the plot labels in Figure 3, 4 and 6.

6 - Eq. (124) should read $[[H_{m_{M+1}},H_{m_M,...m_2,m_1}],\rho]$.

7 - After Eq. (50) it should be specified that $\mathcal H_m\propto\mathcal S_z$ for $m\neq0$.

8 - (Not required) it would be nice (for my personal curiosity) if the authors could comment on remark b) and e).

  • validity: high
  • significance: high
  • originality: good
  • clarity: good
  • formatting: excellent
  • grammar: perfect

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