On the generality of symmetry breaking and dissipative freezing in quantum trajectories

Dear Editor and Referees,

We thank both referees for their thorough comments and reading of the manuscript. Following these comments, we have revised the manuscript substantially and believe it merits publication in Scipost Physics. Below, we first respond to the main criticisms of Referee 2, we then provided itemised responses to the individual comments of the referees. A list of the changes we have made to the manuscript is also provided for the resubmission.

The authors, Joseph Tindall, Dieter Jaksch and Carlos-Sanchez Munoz

Main Criticism from Referee 2 (Lack of precision and originality):

Here we address the referee’s criticism about the originality and precision of the manuscript.

Originality: We disagree that References [24-26] communicated the picture we provide in Section 1 and 2 of our work. All these references considered a single microscopic model and numerically observed dissipative freezing (DF) (with Ref. 24 the only one to provide a mathematical proof and this was specific to the exceptional situation H = L). We instead make no assumptions about the details of the Liouvillians—other than its possession of a strong symmetry—and provide significant mathematical and physical insight into why freezing will emerge. We also identify a direct connection between rigorous mathematical results on ergodicity [40-42] in open systems and dissipative freezing – something which Refs. [24-26] all missed. We did not intend our work to constitute a rigorous mathematical proof of DF. The aim of this paper is, instead, to provide an insightful, comprehensive, mathematical and physical understanding of its origin, complemented with a range of concrete numerical examples. Alongside the intuition and comprehension it provides, such a perspective has allowed us to develop a number of original results without needing to rely on heavy mathematical artillery: exceptions to DF, a lower bound on the timescale on which DF occurs and a computationally efficient method for identifying traceless non-decaying modes in a Liouvillian. In our revised version of the manuscript we have made these points much clearer.

Precision: Both referees comments have significantly helped us improve the precision of the manuscript and the changes made are listed at the end of the document. These changes include mathematical definitions for certain quantities, rectifying grammatical and mathematical errors, avoiding discursive discussions, providing several elaborations when requested and deriving a rigorous lower bound on the time-scale of dissipative freezing – which we demonstrate in our numerics. We believe that these changes have significantly improved the clarity and precision of the manuscript.

Report 1 Responses:

1) The time-derivatives were in place to emphasize that the expression is `steady’ i.e. that the ensemble average of coefficients is 0 and unchanging in the long-time limit. The referee is correct, however, that this is superfluous and $m=0$ is sufficient because if the limit $t \rightarrow \infty$ converges asymptotically for $m=0$ then its derivative must vanish.

2) For a given trajectory (i.e. specific $i$) if freezing has occurred then there is only a single α for which the coefficients $c_{\alpha, \beta}^{(i)}(t)$ are not zero. Hence, for a given $i$, the product of coefficients for two different values of α is zero. We have now made this clear.

3) Yes, we did not intend our definition to imply you cannot have traceless non-decaying modes within a single symmetry subspace. We have renamed these as `inter-sector traceless modes’ in order to avoid suggesting these are the only possible traceless, non-decaying modes that can appear in a Liouvillian’s spectrum.

4) One is always able to introduce a new symmetry subspace to the system by expanding their Hamiltonian and jump operators via a direct sum. Physically this would correspond to considering an additional, independent open system on top of the existing one.

5) We now provide a mathematical definition of this average growth rate.

6) This time average is defined via an average between times $0$ and $\infty$. We have specified this and improved the readability here as the sentence was confusing in its original form.

7) The connection we were implying is that establishing an ergodic theorem with a single trajectory across multiple symmetry sectors is not possible due to the problem being over constrained in analogy with Eq. (10). In the interests of avoiding discursive discussion we have removed the reference to Eq. (10) here.

8) We have introduced explicit formulas for the quantities mentioned at the end of Section 2.3. With the addition of more explicit formulas for the freeze-time and inter-sector spectral gap (see point 10), this should be easier to follow.

9) Yes, the first is a breakdown of freezing despite Eq. (8) holding whilst the second is a scenario in which Eq. (8) is no longer true. Section 2.5 has been reworded to make this clearer.

10 + 11) We have introduced an equation for the freeze time and inter-sector spectral gap. Our expression for the freeze-time now follows from a new section where we derive a lower-bound on the timescale of freezing.

Report 2 Responses (under `Lack of Precision’)

1) The basis of steady states is at least $D$ dimensional, this is now stated explicitly.

2) We have provided additional discussion and references about the relationship between a stochastic unravelling of the GSKL equation and a continuous measurement process in an open system. We note that non-Hermitian jump operators can result in a physical unravelling – for instance via photon counting in the environment.

3) The ‘Traceless’ mode exception completely covers the situation in which the inter-sector spectral gap vanishes, leading to Eq. (10) not being valid and freezing not occurring. The `Similar Symmetry Subspaces’ exception we detail is the only instance we are aware of where Eq. (10) can hold and the system does not undergo dissipative freezing. It is possible other examples where the structure of the two subspaces is sufficiently similar to prevent freezing exist. We have made this explicit, along with the fact we are, however, unaware of any such examples.

4) We thank the referee for spotting this error—we meant to say the projected Hamiltonian has the same form when focussing on the diagonal block where it is non-zero and have now made this mathematically precise, including the part about the unitary transform (change of basis).

1) Introduction. We have mentioned the bound we derive on the freeze-time and improved the wording to make the contributions of the paper to the field clearer.
2) Section 2.1. We have clarified that we mean the basis of steady states is at least D dimensional.
3) Section 2.2: We have expanded on what we mean by ‘generically’ in the definition of dissipative freezing.
Section 2.3: Recovering a Block Diagonal Steady State: The time derivatives have been removed from Eq. (10). The sentence following Eq. (10) has been made clearer. The discussion at the end of this subsection has been made more concise and clear.
4) Section 2.3: Long Products of Matrices: We have mathematically defined the average growth rate of the singular values of the propagation matrices in a given symmetry subspaces.
5) Section 2.3 Ergodicity and Quantum Systems Subject to Repeated Measurements: We have removed the discussion about Eq. (10) in this section to avoid being too discursive. We have mentioned that the time-average in the references is defined as between times $0$ and $\infty$.
6) New Section 2.4: The Timescales of Dissipative Freezing. This is a new section in which we derive a lower bound on the freezing time of trajectories. In this section we have also provided a mathematical definition for the inter-sector spectral gap, which plays a central role in the lower bound.
7) Section 2.5 (Previously Section 2.4): We have emphasized that Eqs. (8) and (10) are maintained in the first case whilst breaking down in the second, making the two exceptions very distinct. We have correlated the breakdown of freezing in terms of the bound derived in Sec 2.4. We have corrected a mistake in Eq. (15), including the unitary transform, and re-worded the sentences following to provide clarity. We have also made it clear that other exceptions involving similar subspaces may exist beyond Eq. (15).
8) Section 3.1: We have provided a new and clearer definition (the qualitative results remain the same as before) of the average freeze-time, connecting it to our new section 2.4 (on the timescales of freezing). We have mathematically defined the types of initial state used in the numerics to be precise.
9) Section 3.2-3.4 Numerical Results. We have redefined the freeze-time and updated the plots based on this (the qualitative physics is unchanged). We have plotted the lower bound on freeze-time in Figs 3c and 4c. We have significantly updated the discussions to reflect the earlier changes in the manuscript (e.g. new section and the earlier definition of quantities like the inter-sector spectral gap). This makes the discussion in this section much more precise.
10) Throughout: Grammatical errors pointed out by the referees have been fixed, included a naming error in the references.