Reply to anonymous report 3

We thank the referee for their detailed critique. The points raised by the referee underline the importance of a discussion on certain concepts in the framework of discrete time crystals (DTCs). As raised by the expert third Referee, there is confusion in the literature on nomenclature and the physics that DTCs entail, which our work rectifies. We are more than happy to summarize the main points of our manuscript once more.

Point 1:

The revised version of the manuscript does not address my main concerns. I have no comments regarding the classical analysis presented in the article, but the quantum part and its relation to quantum discrete time crystals (DTC) are misleading. My specific criticisms are as follows:

The results of the quantum problem considered in the article correspond to a two-mode system with only a few photons (see Fig.2(c)-2(e)). Such a system has nothing to do with spontaneous symmetry breaking in quantum systems and formation of a DTC, where only in the thermodynamic limit states with broken symmetry can live infinitely long. The statement given in the introduction that "We observe that the quantum many-body system forms normal modes [fulfilling (i) and (iii)] with DTC phases that mix over time through quantum fluctuations" pertains to the case of several photons, and not to DTC, and is therefore not true.

Reply 1:

We would like to start by emphasizing that we present a general theoretical treatment for a quantum system with N->infinity modes. Based on the observations in the large system, we then move to present a representative example calculated for a small photon number with two modes. However, the concept of noise-activated dynamics is generally applicable. The fact that the large-amplitude limit (or alternatively the large-network limit) exponentially increases the stability of a time crystalline state is exactly the point that we want to emphasize. In the revised manuscript, we no longer refer to this as the “classical limit” to avoid confusion. Nonetheless, it is clear that such larger amplitudes, or large (coherently coupled) networks, are much easier to fulfill in a “heavy” classical system whose amplitude substantially exceeds quantum and thermal fluctuations.

We have now removed the term ‘classical’ in the following sentences:

• in the last paragraph of section I: “Our treatment highlights that condition (ii) is much easier to fulfill in a classical system, as its large amplitudes are more resilient to fluctuations.”
• in the last paragraph on page 4, “In the limit of high normal modes' amplitudes, the influence of these quantum fluctuations rapidly decreases concomitant with a suppression of activation, cf. Appendix A.”
• in the first paragraph in section VII: “Interestingly, as massive mode populations provide resilience against fluctuation-induced activation between various attractors, systems with large amplitudes have diverging lifetimes. This condition is easy to fulfil in classical systems.”

Point 2:

The authors present quantum results for several photons in the system. Equation (8) correctly describes quantum fluctuations not for several photons, but in the limit of a large number of photons, i.e., in the mean-field limit. In the mean-field limit, quantum fluctuations compared to the values of \alpha_j are negligible besides close vicinity of a bifurcation point, where quantum fluctuations are large and the mean-field description breaks down. In the mean-field limit, thermal fluctuations in the quantum system can be described by equation (8), and for sufficiently high temperatures, it can be expected that they will be similar to the thermal fluctuations in the classical case. That is, it is well known that the Bogoliubov-de Gennes operator in the description of fluctuations around the mean-field state is exactly the same as the Jackobian matrix of (7), see e.g. chapters 6 and 7 in Castin, arXiv:cond-mat/0105058. Therefore, while thermal fluctuations in the quantum and classical dissipative DTCs may be similar, the connection of classical thermal fluctuations with quantum fluctuations for a small number of photons does not seem to be justified.

Reply 2:

Again, we fully agree. The referee echoes our point that truly non-classical phenomena differ from a mean-field approach. However, these phenomena are short-lived in the presence of fluctuations and therefore are not considered in our manuscript. In addition, as the Referee also highlights, spontaneous symmetry breaking required by DTCs manifests only in the large-amplitude/thermodynamic limit. Such massive stationary motion sustains standard quasiparticle excitations which are readily populated by quantum or thermal fluctuations. Note that our Fig. 2 in the low-photon limit is used to demonstrate the rapid convergence from quantum to the semiclassical limit in driven systems. There, even for cavities with approximately 2 photons on average, the mean-field treatment yields a good description.

Point 3:

The quantum tunneling between symmetry broken states of the two-mode system with a few photons is confused with the problem of pre-thermalization in many-body systems in the presence of many modes that couple to states corresponding to time crystals. Quantum tunneling decays exponentially quickly in the thermodynamic limit, and is not related to the pre-thermalization effect discussed in discrete time crystals (see Ref.[3]), which may be present even, and perhaps especially, in the thermodynamic limit.

Reply 3:

As we write in reply to the third referee, we agree that the terminology of ‘prethermalization’ is confusing and not necessary in the present case. We are happy to avoid it. In the revised manuscript, we exchanged the term with ‘DTC lifetime’.

The mechanism underlying spontaneous symmetry breaking in quantum systems was already discussed in our reply to the referee’s question 1 in the first round of review, see our response from September 7, 2022. To summarize, indeed, large-amplitude or thermodynamic systems become more resilient to quantum fluctuations with size.

We would like to stress that even though quantum tunneling may be suppressed in the thermodynamic limit, activation via noise remains the main reason why two-level fluctuators appear ubiquitously in physics. In many quantum engineering disciplines, the influence of spurious two-level systems on a single engineered quantum degree of freedom has been identified as a fundamental and impactful source of decoherence that proves immensely hard to mitigate.

Point 4:

The system considered in the article is open. The term "dissipative time crystals" should appear in the title.

Reply 4:

Again, this point was addressed in our reply to the referee’s question 3 in the first round of review, see our response from September 7, 2022. As we explain there and in the manuscript, dissipation as such has no important consequences in our system (apart from slightly shifting the boundaries of the phase diagram). It is the fluctuations that are crucial for driving the population of the Bogoliubov modes and for the restoration of symmetry over long timescales. We emphasize that such fluctuations are also present in closed systems without dissipation. Therefore, we prefer to not distinguish between dissipative and non-dissipative DTCs. Such a distinction leads to a misleading notion of different ‘classes’ of DTCs that, upon close scrutiny, are not fundamentally different. By using the term ‘fluctuation’ instead of ‘dissipation’, we believe our paper is clearer and more valuable to the readership.

Point 5:

The second paragraph on page 3 and Appendix B are misleading in the context of time crystals in many-body-localized (MBL) spin systems. Discrete time crystals in MBL systems are represented by pure states, and the thermodynamic limit is necessary because then they live infinitely long. Meanwhile, the authors in Appendix B consider mixed states composed of different states with broken symmetry, in which symmetry breaking and DTC will not be visible. It is like conducting many DTC experiments with pure states in the Google sycamore quantum computer Ref.[21] and then averaging the results.

Reply 5:

Limiting the discussion to coherent states without superpositions leads to a semiclassical picture without true quantum features. This fully coincides with our narrative that only the large-amplitude regime (or the large-network regime) can have exponential stability in the DTC phase. Such systems can be well described by a deterministic classical theory.

Our motivation for adding appendix B after the previous round of reports was to allow for a unifying discussion of the diverse possible quantum regimes. This appears necessary, e.g. in light of the referee’s first question in their first report. If preferred, we will be happy to remove appendix B again, as it is not important for our main message.

The Google experiment shows that a delocalized quantum state forms within the coherence time of the device, while still having time to respond to a drive with a slower rate. This is a nice demonstration of the spatial lifetime of the extended state in the device, and its ability to respond as a subharmonic Floquet state. Note, however, that under the requirements of DTCs, such a Floquet state should be very massive (thermodynamic limit), and only then fulfill criterion (ii).

In coupled bosonic cavities, we can understand that each photon in a normal mode of our system realizes such a quantum state. The large amplitude limit leads to a similar requirement as the thermodynamic limit, and the required ‘DTC-resilience’. As such, we fail to see how the Google experiment with a small system and short-lived excitations generates a more convincing proof of DTCs than a semiclassical large system that undergoes proper symmetry breaking.

Point 6:

In the last paragraph of Sec.II, the authors write that "Opening the system to weak dissipation channels does not impact the stability diagram in a radical way." This is true for the system considered in the article, but it is not generally true for discrete time crystals in closed systems. In MBL spin systems, introducing dissipation kills MBL and DTC, see Lazarides and Moessner, PRB 95, 195135 (2017). Therefore, once again, the authors should emphasize in the title and in the article itself that they are considering a dissipative version of discrete time crystals.

Reply 6:

We replaced this sentence with a specific reference to our system: “Opening our system to weak dissipation channels does not impact the stability diagram in a radical way." Regarding the terminology of dissipative DTCs, please see point 4 above.

Point 7:

It is not clear to me what Hamiltonian the authors are using in (6) to find stationary states. Is it the RWA Hamiltonian (2) or the exact Hamiltonian (1)? If it is the latter, what do they mean by "stationary \rho_s," and at what time t are the results presented in Fig.2?

Reply 7:

We analyze the stationary states of the RWA Hamiltonian Eq. (2) with dissipation. To clarify this, we now use H for the exact Hamiltonian (1) and bar{H} for the RWA Hamiltonian.

Reply to report by Prof. M. Dykman

We are grateful for the work invested by Prof. Dykman in reading our manuscript and in providing constructive criticism, which is highly appreciated. The referee’s comments indeed helped us to improve the clarity of the text.

Point 1:

I would downplay references to prethermalization. The term is used in the meaning that is somewhat different from what is frequently implied when referring to prethermalization. In particular, in paragraph 2 below Eq.(8), it is not clear what is meant by the prethermal regime. It is a cosmetic change, but worth clarifying.

Reply 1:

This is a valuable point that we are happy to implement.

Changes: we now use the terminology ‘DTC lifetime’ instead of ‘prethermalization’ to indicate the finite timescale over which a symmetry-broken state is typically stable.

Point 2:

Two paragraphs below Eq.(8) the reference to “suppression of activation” could be read as if in the classical regime activation is suppressed, whereas it is meant that activation is suppressed in the regime of large vibration amplitudes. This refers also to the statement “classical systems have diverging prethermal timescales, making them superior to their quantum counterparts as DTCs.”: what is meant here, I think, is large vibration amplitudes rather than classical fluctuations, but the text is somewhat ambiguous.

Reply 2:

We agree that our previous text was not sufficiently clear. The referee is entirely correct that it is not the classical limit in the accepted sense, but the large amplitudes that are decisive for the phenomenology we discuss.

We now avoid the term ‘classical’ in this context and explicitly refer to large amplitudes.

Point 3:

I would word more carefully the discussion of the possibility of a time crystal in the coherent regime for a closed system in the thermodynamic limit. A quantum phase transition to the corresponding state has been described in the literature. I believe the authors mean that their condition (ii) does not hold, since the concept of “stability” does not apply in a coherent regime.

Reply 3:

We fully agree with the Referee’s interpretation of our intention. The description in the literature of the closed system quantum phase transition relies on the assumption that a macroscopic quantum state may form and respond subharmonically to a drive w/o induced coupling to any other many-body state. However, AC stark shifts and power-spectral broadening (and any higher order Floquet terms) induced by the drive will effectively couple other states in the system, leading to an effective dissipation channel as well as the propensity of the closed system to heat up to infinite temperatures. Instead, we adopted an effective open system description. This is exactly the reason that we believe that the criterion (ii) can only manifest in a ‘massive’ limit, and benefits from a standard statistical physics interpretation.

We slightly rephrased the corresponding paragraph.

Point 4:

Another term to straighten out is “fluctuations forming normal modes”: the authors are talking about fluctuations of normal modes, whereas the modes themselves are formed dynamically.

Reply 4:

We now see that our formulation here was not well chosen. In the revised manuscript, we draw a distinction between the excitation modes around stationary states and the fluctuations that drive them.

“For coupled resonators, the displacements away from the stable states are themselves coupled and form normal modes (excitation quasiparticles).”

We replaced the term ‘fluctuations’ by ‘excitations’ throughout the text where it pertains to the fluctuation-driven displacement away from the stationary states.

It would be good to explain the advantageous feature of studying fluctuations of symmetric and antisymmetric combinations of the displacements from the stable states of two oscillators rather than studying such fluctuations for each oscillator separately.

We now write after Eq. (10): “The advantage of this procedure is that it isolates the spectral components of the excitations, and allows us to track the individual resonances in the presence of aliasing.”

The paper would benefit if a more careful comparison with the previous work was done. In particular, the switching between coexisting vibrational states is well-understood for a single oscillator and also for weakly coupled multiple oscillators in the quantum and classical regime, see Ref. 11 and papers cited therein. I would also compare Eq. (7) with Eq.(11) in this paper. The problem of the spectra for single oscillators was addressed in PRA 83, 052115 (2011).

As stated by the referee, Ref. [11] studies the weakly coupled oscillator network. In the mean-field limit, Eq. (11) in that paper describes a stochastic Langevin-like dynamics for the position and momenta of the individual oscillators. This equation is indeed analogous to Eq. (7) (written in terms of the expectation value of the mode annihilation operator) in our work. In paragraph 2 on page 4, where activation between solutions is discussed, we now emphasize “Such fluctuation-activated mixing was studied for individual oscillators and for weakly coupled systems, see e.g. Ref [11] and references therein. The related problem of quantum heating at zero temperature was addressed in Ref. [PRA 83, 052115 (2011)]”