Non-stationary coherent quantum many-body dynamics through dissipation

The assumption that quantum systems relax to a stationary state in the long-time limit underpins statistical physics and much of our intuitive understanding of scientific phenomena. For isolated systems this follows from the eigenstate thermalization hypothesis. When an environment is present the expectation is that all of phase space is explored, eventually leading to stationarity. Notable exceptions are decoherence-free subspaces that have important implications for quantum technologies and have so far only been studied for systems with a few degrees of freedom. Here we identify simple and generic conditions for dissipation to prevent a quantum many-body system from ever reaching a stationary state. We go beyond dissipative quantum state engineering approaches towards controllable long-time non-stationarity typically associated with macroscopic complex systems. This coherent and oscillatory evolution constitutes a dissipative version of a quantum time crystal. We discuss the possibility of engineering such complex dynamics with fermionic ultracold atoms in optical lattices.

Referee 1 raises the question of originality of the results, pointing at several papers likewise concerned with time-dependent decoherence free subspaces. I went through some of these references (which I was likely not aware of) and based on that I agree that the mathematical structure reported by the authors was recognized previously. However, in all of these works the focus is on small quantum systems, instead of many-body systems. The examples given there appear rather artificial, if present at all. In the ``worst case', the present work may be seen as leveraging these works to the many-body realm, and recognizing that the dynamical symmetries needed for this setting to work are actually realized in several interesting and relevant many-body Hamiltonians, in terms giving these symmetries even a new physical meaning. In summary, I think that the proper contextualization sometimes may just as valuable as the pure mathematical construction, and this is the actual advance of the present work. Of course, this is a personal view, but there are many examples in physics (e.g. the Berry phase was discovered in optics way before Berry found it, but he figured out the broad scope and applicability of the concept). Obviously, I still suggest that the authors properly account for the previous work.
Here are some more specific comments and concerns, which revolve around the robustness of their effect in the many-body context (so answers are important to strengthen my own reasoning on novelty and contextualization): 1. Realization of the fermion model: Why is it necessary to immerse the fermion system into a bath of bosons? Is the spontaneous emission from the lattice drive laser not sufficient to induce local dephasing? I.e. should one use less far detuned lattices? This is an ubiquitous effect present in any optical lattice (see H. Pichler, A. J. Daley, and P. Zoller, Phys. Rev. A 82, 063605 (2010)): So, why have experiments not seen the coherent oscillations so far? 2. More generally: What is the requirement on the strength of dephasing: Does it have to be much larger than the other scales in the Hamiltonian? Does one need a gap (minimal distance from the imaginary axis in their Supplementary Fig. 1? (Which I do not see, upon increasing the lattice size). 3. How big are the decoherence free subspaces in the context of the one-dimensional examples for thermodynamically large systems? The background of this question is whether, in case they are extensively large (number of states contains scales with system size --seems compatible with their sub-n^2 scaling?), then there should be thermalization/ergodic dynamics be going on in either the subspace. Would that overwrite their effect or is it robust under such subspace ergodicity? 4. In the presence of dephasing (more generally: hermitian jump operators), there is always the fully mixed state stationary solution, which is annihilated by the jump operators and commutes with any Hamiltonian. Usually, this fixed point is attractive in a large system. The author however propose a generalized Gibbs ensemble as the stationary solution. Why, in this light? What about the unit matrix stationary solution? 5. Physical picture: While I follow their mathematical construction, I stumbled over the following (seeming?) contradiction, which the authors could help clarify and improve their physical picture (honestly, I find the high-level description of the effect starting in line 45 rather confusing than enlightening; in particular, what means "symmetry protecting"? symmetry preserving?): If I understand it correctly, there are stationary states in the problem (termed \rho_\infty), but certain observables show persistent oscillations. Usually, one would argue that, given a stationary state density matrix, any correlation function evaluated on it is stationary as well (depends only on the difference in times between the observables, e.g. <\psi^\dag (t) \psi (t') > is a function of t-t' alone). In their case, there is a forward in time evolution for some observables despite a stationary state. How is this understood more intuitively? Or conversely, what is the meaning of the stationary state \rho_\infty in this context? 6. Line 121, what means "n\to \infty" in this context? Obviously the filling/density is bounded.

Reply to Referees of "Non-stationary coherent quantum many-body dynamics through dissipation: Revision 1"
B. Buca, J. Tindall, D. Jaksch 1 Reply to Referee 1 Referee 1: "The authors sa sfactory answered all my ques ons and made appropriate correc ons in the manuscript. The only remaining point on which we disagree is calling the dark Hamiltonian non-Hermi an, (as they also explain) this generator is Hermi an under a proper iden ca on of the space on which it acts. This is a minor point, and in the new version, it is also not featured prominently.
I recommend that the ar cle is published in its current version." We thank the referee again for their very useful and interes ng remarks, sugges ons and ques ons, as well as their posi ve appraisal of our manuscript and the recommenda on to publish our ar cle.

Reply to Referee 3
We thank the referee for their interes ng ques ons and useful comments. We address the concerns raised on a point-by-point basis.

Referee 3:
"The authors present a concept of me-dependent decoherence free subspaces and apply it to driven open quantum many body systems, to show in several examples based on spin models and the fermionic Hubbard model in one dimension that as a physical consequence, certain observables show persistent oscilla ons. This goes beyond the previously established concept of dark states in driven open many-body systems, where the decoherence free subspace is typically me independent (one reason being that it is typically one-dimensional, such that any me evolu on would be trivial).
I nd this paper truly inspiring and I have no doubt that the conclusions are correct, since all general arguments are complemented with concrete numerical simula ons in one-dimensional systems. In par cular, I judge the step beyond the previous dark state scenario an important one; one reason being that it may lead to a rather robust scenario that may be ubiquitous in Hamiltonian quantum systems with strong symmetries, once coupled to (quite natural) dissipa on channels, as illustrated in the Hubbard model example (but see my ques ons below). In this light, if my concerns below can be addressed, I would recommend publica on in Nature Communica ons.
Referee 1 raises the ques on of originality of the results, poin ng at several papers likewise concerned with me-dependent decoherence free subspaces. I went through some of these references (which I was likely not aware of) and based on that I agree that the mathema cal structure reported by the authors was recognized previously. However, in all of these works the focus is on small quantum systems, instead of many-body systems. The examples given there appear rather ar cial, if present at all. In the "worst case', the present work may be seen as leveraging these works to the many-body realm, and recognizing that the dynamical symmetries needed for this se ng to work are actually realized in several interes ng and relevant many-body Hamiltonians, in terms giving these symmetries even a new physical meaning. In summary, I think that the proper contextualiza on some mes may just as valuable as the pure mathema cal construc on, and this is the actual advance of the present work. Of course, this is a personal view, but there are many examples in physics (e.g. the Berry phase was discovered in op cs way before Berry found it, but he gured out the broad scope and applicability of the concept). Obviously, I s ll suggest that the authors properly account for the previous work." We thank the referee for their posi ve appraisal of the manuscript. We now include cita ons to all the previous work men oned by both of the referees. We were also not aware of some of the work men oned. We now address the points raised.

1.
Referee 3: " 1. Realiza on of the fermion model: Why is it necessary to immerse the fermion system into a bath of bosons? Is the spontaneous emission from the la ce drive laser not su cient to induce local dephasing? I.e. should one use less far detuned la ces? This is an ubiquitous e ect present in any op cal la ce (see H. Pichler, A. J. Daley, and P. Zoller, Phys. Rev. A 82, 063605 (2010)): So, why have experiments not seen the coherent oscilla ons so far? " In the limit of red-detuning, provided we can neglect the o -diagonal terms in the dissipator, the dominant e ect of the sca ering is to return the atoms to the lowest band, and the corresponding e ect of the la ce dissipa on is to introduce pure local on-site dephasing (c.f. eq. (28) of the reference) as stated by the referee.
The spin-1/2 fermions in our ultra-cold atom se ng are implemented as two atomic hyper ne levels. The incoherent light sca ering will, in general, dis nguish between the two spins. This would result in two Lindblad operators per site L i,↑ = n i,↑ and L i,↓ = n i,↓ , where n i,↕ is the number of spin-up, spin-down fermions on site i. In order to respect the condi on below line 74 of the main text of our manuscript, we require that the Lindblad operators commute with the total spin raising operator. This might be achievable in some experiments by ne-tuning the la ce laser parameters but will in general not be the case.
In contrast, in the setup described in the Supplementary Material the BEC interacts via spin-independent density-density interac on. This guarantees that the decoherence will not dis nguish between spin-up and down. This di erence is subtle, but crucial. Physically, the rst case corresponds to a measurement (classical random process) that dis nguishes between the di erent spin states. The second case corresponds to a measurement in a spin-agnos c way.

2.
Referee 3: " 2. More generally: What is the requirement on the strength of dephasing: Does it have to be much larger than the other scales in the Hamiltonian? Does one need a gap (minimal distance from the imaginary axis in their Supplementary Fig. 1? (Which I do not see, upon increasing the la ce size)." There is no requirement on the strength of the dephasing within our model. We only need that other sources of dissipa on that do not respect the symmetry requirement are small compared to it. What happens when there are addi onal large sources of such unwanted dissipa on is an interes ng open ques on. We plan to study this in the near future -one may imagine the possibility of a dissipa ve phase transi on depending on the ra o of the strengths of the di erent types of dissipa on. This ques on is related to the one of metastability that we have men oned in the concluding paragraphs.
The strength of the dissipa on does control the size of the gap and thus the me it takes for the system to reach the non-sta onary in nite-me limit. We have added a sentence to the last paragraph of p. 3 to make this clearer: "The strength of the system environment coupling determines the me for the transient dynamics to decay and coherent, oscillatory behaviour to appear." Supplementary Fig. 1 shows the spectrum of the XXZ spin ring with a single loss term. With increasing system size, the gap closes and new frequencies enter into the long-me dynamics. If these are dense and incommensurate, eigenstate dephasing (in the closed system sense) is possible and this could happen for the XXZ spin ring. An analy cal and/or numerical analysis of this thermodynamic limit is an interes ng open ques on. We note that in contrast to the XXZ spin ring we do not see new frequencies entering the long-me oscillatory dynamics in the Hubbard model discussed in the main text.
We now comment on the possibility of eigenstate dephasing in the XXZ spin ring in the Supplementary Material by including the sentence: "With increasing system size, new frequencies enter into the long-me dynamics. If these are dense and incommensurate, eigenstate dephasing does become a possibility."

3.
Referee 3: "3. How big are the decoherence free subspaces in the context of the one-dimensional examples for thermodynamically large systems? The background of this ques on is whether, in case they are extensively large (number of states contains scales with system size -seems compa ble with their sub − n 2 scaling?), then there should be thermaliza on/ergodic dynamics be going on in either the subspace. Would that overwrite their e ect or is it robust under such subspace ergodicity?" We es mate the size of the subspaces for the Hubbard model example is M 2 − M. There can be thermaliza on/ergodic dynamics within each subspace -that is the crucial di erence between this example and a usual decoherence free subspace composed of pure states. Even when each individual subspace has relaxed by itself (mathema cally, the eigenspaces contain only mixed states), remarkably, the joint dynamics will s ll be non-sta onary. This is because the subspaces all have commensurate frequencies and thus they can never dephase each other in the closed system sense (destruc ve interference due to incommensurate frequencies).
In order to make it clearer that the e ect is robust regardless of the thermaliza on in each individual subspace we have changed the sentence on p. 3 of the main text that said: "The long-me dynamics is then periodic with period 2π/λ " to: "The equidistance of the spectrum ensures the long-me dynamics is periodic, with period 2π/λ and the system does not relax to sta onarity."

4.
Referee 3: "4. In the presence of dephasing (more generally: hermi an jump operators), there is always the fully mixed state sta onary solu on, which is annihilated by the jump operators and commutes with any Hamiltonian. Usually, this xed point is a rac ve in a large system. The author however propose a generalized Gibbs ensemble as the sta onary solu on. Why, in this light? What about the unit matrix sta onary solu on?" The referee is correct. The unit matrix is a sta onary state which corresponds to se ng β 0 = β 1 = β 2 = 0 in our parametriza on of sta onary states. The important point is that this is not the only sta onary point in our system. The symmetry proper es discussed in the manuscript require the quan es N, S Z , S + S − to be conserved during the evolu on. This in general prevents the dynamics from reaching the unit matrix sta onary state.
To emphasize the importance of these quan es being conserved we have added the following sentence: "The quan es N, S Z , S + S − are conserved during the me evolu on."

5.
Referee 3: "5. Physical picture: While I follow their mathema cal construc on, I stumbled over the following (seeming?) contradic on, which the authors could help clarify and improve their physical picture (honestly, I nd the high-level descrip on of the e ect star ng in line 45 rather confusing than enlightening; in par cular, what means "symmetry protec ng"? symmetry preserving?): If I understand it correctly, there are sta onary states in the problem (termed ρ ∞ ), but certain observables show persistent oscilla ons. Usually, one would argue that, given a sta onary state density matrix, any correla on func on evaluated on it is sta onary as well (depends only on the di erence in mes between the observables, e.g. < ψ † (t)ψ(t ′ ) > is a func on of t-t' alone). In their case, there is a forward in me evolu on for some observables despite a sta onary state. How is this understood more intui vely? Or conversely, what is the meaning of the sta onary state ρ ∞ in this context?" We agree with the referee that the term "symmetry protec ng" is confusing and have changed the wording to "symmetry preserving". This phrase makes the symmetric nature of the dissipa on clearer.
The referee is correct, our system features mul ple sta onary states ρ mm . Importantly we also nd eigenmodes ρ nm with n ̸ = m that have purely imaginary eigenvalues and hence do not damp out in the long-me limit. If these eigenmodes are contained in the ini al state the system will never reach a sta onary state. Instead, the eigenmodes with purely imaginary eigenvalues will con nuously oscillate. They are the physical origin of the oscilla ons in single-me observables seen in our work.
The two-me correla on func ons men oned by the referee may also show interes ng behaviour, which we will study in the future.
We realize that our nota on ρ ∞ may have given the wrong impression that this is the only state that can be reached a er a long me. In order to rec fy this we have now uni ed the nota on of eigenmodes and renamed ρ ∞ → ρ 00 .

6.
Referee 3: " 6. Line 121, what means "n → ∞" in this context? Obviously the lling/density is bounded." We thank the referee for poin ng this typo. We have removed it and replaced it with ∀i, j.