Population Dynamics of Schrödinger Cats

Dear Dr. Beenakker,

Thank you for your review of our manuscript. We also thank the referees for their suggestions for improving the manuscript. We have expanded various sections of our manuscript in response to the referees comments; our replies to specific referee comments and the specific modifications to our manuscript accompanying them are detailed below.

With best regards,
Foster Thompson and Alex Kamenev

Response to Referee 1:

1: The constraints of the action are not known (to us) to be reducible to a symmetry, but are instead imposed by a combination of the dark state FDT-like condition Eq. (35) and the requirement that the stationary state be fluctuationless, which is encoded by Eq. (39). A paragraph following Eq. (40) has been added to clarify this point.

2: The cubic vertices derive from both coherent and incoherent processes $\sim\hat a^\dagger\hat a\hat a$ in which the particle number on one side of the density matrix is changed. In this sense, they can be loosely interpreted as branching and fusion processes on one side of the density matrix at a time. Some comments were added to section 4 in the paragraph following Eq. 41 and the caption of Fig. 6 to this point.

3: The the existence of large-N theories of either the classical or quantum population models is an interesting question which, to our knowledge, has not been thoroughly addressed. It is believed that absorbing state phase transitions in classical population models with multiple species generally fall into the directed percolation universality class (however there may be multi-critical points where the scaling is modified). How this generalizes to the quantum setting and whether or not it admits a large-N description may be interesting questions for future work. We have expanded on this point in our conclusion section.

Response to Referee 2:

1 and 2: We thank the referee for bringing these references to our attention. We expect the nature of our phase transition to be similar to classical directed percolation: the phase transition should persist in all dimensions. Above the critical dimension $d\geq4$ the scaling will become mean-field (but may occur at finite critical ratio due to fluctuations, as shown by the references the referee cites). Higher than cubic vertices may become relevant for $d<3$ which may modify the scaling, however just as in DP the cubic vertices will remain be the most relevant interactions and determine the universality class. A paragraph directly preceding section 4.1 was added to clarify these points.

3: Axes and axis labels have been included in phase portrait plots.

Response to Referee 3:

1: It is a good point that many known Lindbladian theories with both Hamiltonian and incoherent terms display critical behavior in some known classical (equilibrium or non-equilibrium) universality class. The explicit breaking of weak symmetry does not guarantee a new universality class (for one, the a symmetry-breaking perturbation may turn out to be irrelevant, leading to its recovery in the IR; it also may turn out that any new terms permitted by the absence of symmetry do not modify the critical scaling). It nevertheless remains a possibility that the symmetry-breaking perturbation is relevant and drives the theory to a new universality class; we argue that the theory we present is one such example. We agree that, as it is currently written, our discussion of this point does clearly convey this point and thus may be confusing or misleading. The part of the introduction discussing this point has been rewritten to better reflect past results and contextualize our work in relation to them.

2: The discussion in section 3.2 has been expanded and several remarks added throughout to clarify the various points raised. A footnote has been added demonstrating the validity of Eq. (35) for general interactions.

3: The quantum scaling does not assume nonzero $\alpha$. That is, even at the critical point one finds that the scaling dimensions of the classical $\phi$ and quantum $\chi$ fields are equal. The exact value $-d/2$ is only the bare scaling dimension and does receive perturbative corrections at the critical point. A comment has been added to section 4 to emphasize this point.
It is unclear what relation the RG relevance of the quantum processes (coherent or incoherent) to the classical population dynamics has to the quantum scaling. As noted in the text, the Reggeon field theory describing the completely classical directed percolation transition has the same property, so it is not unique to quantum models.
The relationship of dark state transitions to entanglement is an interesting question and we agree that it is something deserving of further study. A paragraph has been added in the conclusion section briefly discussing this point.

4: The classical directed percolation action cannot be retrieved as a limit of the quantum population action. The physical meaning of the fundamental fields is different: the classical field $n$ is the population number while the quantum action is written in terms of the complex coherent state field $\phi\sim\sqrt n$; this point is discussed at the end of section 3.2.

5: Limit cycles in our theory do not occur in full phases, but rather occur at the separation between the active and dark phases. To clarify this, and other details about the phase diagram, we have modified the language at several places in section 4.2 and 4.3 and also included an additional figure with examples of mean-field phase diagrams.