The role of fluctuations in quantum and classical time crystals

Report 1

We are grateful for the time and effort the referee invested in the second round of review. Some of the comments of the referee clearly helped us to strengthen our manuscript (see point 2). Some critique remains regarding the quantum-to-classical divide, which we believe to arise from a miscommunication on our part (see point 1). In the following reply, we attempt to clarify this matter.

The referee deems our previous revisions and explanations insufficient to warrant publication in the present form, mentioning two issues:

Point 1:

The paper conveys the message that time-crystal phases are classical in character and that quantum effects are not particularly relevant. Some quotes from the most recent version of the paper: 'demonstrating unambiguously that classical and quantum DTC share the same basic properties' and 'We conjecture that there is no fundamental distinction between dissipative classical and quantum time-crystalline phases in this type of system.'

There is a significant amount of literature demonstrating the unique properties of quantum time crystals (and realizing them experimentally) – see e.g. Refs [1-20]. The models considered in these references are not similar to the referee's Hamiltonian as they often take the form of spin models which have no classical analogue. Thus, to me, the author's Hamiltonian is not so general and their conclusions are more limited than the paper suggests.

Reply 1:

We do indeed arrive at the conclusion that quantum effects (i.e. non-classical effects) are unimportant for DTCs. We show this on the example of two coupled KPOs. Nevertheless, based on general textbook principles of quantum mechanics, it appears evident to us that the same should be true for any realistic system.

When we speak of quantum effects, we explicitly mean only effects that do not arise in classical systems, such as coherent superposition. Furthermore, our arguments only concern the long-time limit which is relevant for DTCs, and experimentally realizable systems with a finite degree of coupling to an environment. Basically, what we claim is simply that quantum coherence in an open system cannot be preserved over infinite timescales. Decoherence, in turn, leads to statistical mixing between states with different symmetries, and therefore to a restoration of the overall symmetry. A system with a restored symmetry does not count as a time crystal. In our understanding, these statements form the foundations of statistical physics and are not controversial at all.

We are aware that the time crystal community often cites the idea of a infinitely large, yet perfectly closed system composed out of perfectly uncoupled subsystems, such as spins. It is valid to speculate about such a system, and we have now revised our manuscript to mention this idea. However, we limit our argumentation to partially open systems, because each realistic system is of finite size and has a finite coupling to the outside world (especially when it is driven). The experiments that the referee cites are clearly open systems and suffer from loss of coherence over long timescales. They show time-translation symmetry breaking, which is fascinating, but do not fulfil criterion (ii).

We emphasize that it is not relevant if an experiment was conducted with spins or an oscillator. The mean-field behavior of a single spin can be described entirely with classical mechanics (see e.g. the second chapter in "Principles of Magnetic Resonance" by C. P. Slichter). Any non-classical effect arising in these spin systems, such as a quantum superposition, vanishes beyond a very short timescale.

Changes 1:

Throughout the manuscript, we emphasized that our arguments pertain to systems subject to fluctuations. Furthermore, we revised our conclusion section and formulated our claim in a more specific way to avoid misunderstandings. In appendix B, we provide the relevant quantum notation to support our discussion of coherent quantum states.

Point 2:

The author's analysis of the Hamiltonian for general N is confusing and unclear. The Hamiltonian in Eq. (1) is, in general, difficult to solve as it contains up to quartic terms. It is thus unclear to me:

i) In which parameter regimes (i.e. how weak does V_{j} need to be?) the results of Section II and the conclusions of the paper are valid.

ii) Which parts of the authors' analysis on Eq. (1) are original versus which parts of their analysis stem from previous literature or the classical limit of the Hamiltonian (the authors suggest Fig. 1b was calculated from 'the parameters of the classical system' – I am confused as to what this means and how it relates to Eq. (1))

iii) At which point the author's analyze the system for N > 2 in the dissipative regime. The author's reply suggest their discussion includes the 'general case of N oscillators' yet I can only see meaningful quantitative analysis for N = 2 oscillators.

Reply 2:

We gratefully take this opportunity to clarify these questions.

i) The theory is valid for strong coupling, while the rest of the terms are small, i.e., small detuning, small nonlinearity, parametric driving not much larger than the threshold value, and small damping. These are the usual constraints of perturbation theory, and the regime relevant for most experiments. This is the most stringent limitation beyond the fact that we only study the long-time solutions of open systems.

We added after Eq. (2) the explanation: "The RWA is equivalent to the lowest-order Floquet expansion, and relies on the fact that the corrections to the linear mode basis are nondegenerate and small [51]."

ii) There are several steps taken in the paper. We elucidate the origin and novelty of each step:

a. The standard rotating wave approximation is taken on the Eq. (1) in the most common and standard way – appearing often in the literature.

b. On top of the rotating Hamiltonian, we move to the normal mode basis – a common transformation that appears often in the literature.

c. In the normal mode picture, we describe the appearing Hamiltonian terms. We then argue, based on standard perturbation theory, which terms vanish to lowest order in the relevant parameter regimes (I-III in the manuscript) – this is also a standard approach but its application to the KPO network is new.

d. The semiclassical treatment plus study of fluctuations is a standard technique, which we apply to the KPO network.

e. The stationary solutions of the experimental system were characterized in a previous study, see Ref. [50]. Figure 2(b) is a reproduction of the theory result to serve as an overview for the reader.

iii) It is correct that we present quantitative results only for the case of N=2, because these are the results that we can compare to an experiment. However, Eqs. (2) and (6) provide a recipe for treating systems for arbitrary N. The occurrence of normal mode in the limit of a damped harmonic oscillator is obvious, and Eq. (4) describes the effects of the parametric drive in the normal-mode basis. As long as the nonlinearity is weak enough, the individual modes are essentially uncoupled and their behavior follows directly from that of a single parametric oscillator, which was treated in Ref. [46] among many other works. ---

Report 2

In our previous reply to the referee's first comment, we provided detailed arguments as to the nature of the symmetry breaking and waveform collapse in quantum systems. These arguments are based on textbook quantum mechanics and on the derivations provided in our own manuscript. The referee did not enter a scientific debate at all, accusing us instead of lacking "mathematical accuracy". The short and disparaging referee report does not contain any new information and is not constructive in a scientific environment. We hope the referee is willing to reenter a fact-based, constructive exchange with us.

For the benefit of the Referee, we added section B in the appendix to clarify the mathematical notation of quantum states. We will be happy to provide further clarification upon request.