The role of fluctuations in quantum and classical time crystals

Report 1:

We thank the referee for reading our manuscript and appreciate their feedback and questions. Below please find our reply to the points raised in the report.

1) The referee writes:

It would be helpful for the authors to discuss more thoroughly [...] the motivation behind the model and its relation to previous works by the authors such as Phys. Rev. Lett. 123 124301 2019 and Phys. Rev. Lett. 123 173601 2019.

Our reply:

Period doubling, a key feature for time crystals, is known for nonlinear parametric oscillators since a long time, see Refs. [29,30,31]. In PRL 123 124301 we showed that the period doubling effect persists for a coupled classical system and discussed different coupling strengths and their importance w.r.t. many-body effects in classical time crystals. However, with regards to time crystals, we were often confronted with the viewpoint that the classical and quantum Hamiltonian should lead to intrinsically different phenomenology. The model we present in Eq. (1) is the quantum version of the coupled equations used in PRL 123, 124301 (2019), and an extension (from one to many) of the model in PRL 123, 173601 (2019). Our goal here is really to present a unifying picture that demonstrates the analogy between all of these works.

Changes:

Following the referee’s recommendation, we highlight this relationship more clearly in the manuscript after Eq. (6).

2) The referee writes:

How Fig. 1b is produced for both $\gamma=0$ and $\gamma \neq 0$, including what the actual scale for the $x$ and $y$-axes is in this Figure and the explicit relationships between the $\tilde{\Delta}_k$ , $\tilde{G}_{lk}$ and the parameters of the original Hamiltonian as well as the dependence of the lobe structure on these parameters. Presumably this figure is dependent on the explicit geometry used in Fig. 2a)?

Our reply:

We thank the referee for pointing out this confusion. Fig 1b is a schematic representation of the instability lobes of coupled parametric oscillators. The lobe of a single mode can be calculated in a straightforward fashion from Ref. [49] and [50] for the non-dissipative and dissipative cases, respectively. Considering the case of coupled linear oscillators, the system forms normal modes subject to parametric driving. As the resonance frequencies corresponding to the normal modes differ, the instability lobes are split apart and are centered at separate resonance frequencies. Fig 1b shows an example for identical oscillators. As the normal mode frequencies depend on the coupling $J_{jk}$, this figure also depends on the geometry/the coupling matrix $J_{jk}$. We did not calculate the explicit relationships between the $\tilde{\Delta}_k$ , $\tilde{G}_{lk}$ but rather used the parameters of the classical system to obtain these figures. In Appendix A, we discuss the link between the parameters of the classical and quantum model.

Changes:

In the figure caption, we clarified that the Fig. 1b is a schematic phase diagram.

3. The referee writes:

The authors, however, do need to specify what the parameter $\kappa$ is in the caption of Fig. 2 and what value of $\gamma$ has been used here.

Our response:

We thank the referee for calling our attention to this typo. Indeed, $\kappa$ should be replaced by $\gamma$.

Changes:

We fixed the typo.

4. The referee writes:

The authors then proceed to describe an experimental implementation of their Hamiltonian with a classical setup, stating "the resonator network can therefore be modelled by Eq. (1)". Change 4) Here, the authors should clarify that they mean "modelled by the classical limit of Eq. (1)".

Changes:

We have incorporated this suggestion.

5. The referee writes:

The results of their experiment then lead the authors to conclude that the qualitative features of the quantum model are captured by this classical experiment: they finish with the line that ‘there is no fundamental distinction between classical and quantum time-crystalline phases’. In my opinion, such a conclusion is far too strong given that the authors only analyse a specific quantum Hamiltonian, with specific parameters and only $N=2$ oscillators in the fully quantum regime. The dependence of the system on $N$, the oscillator geometry and the strength of the nonlinearity $V$ are not included here. The conditions for a DTC include `sufficiently long-range correlations’ (the authors words), something which cannot be analysed in the authors’ work due to the small system sizes involved. Change 5) I would therefore suggest that the authors significantly temper their conclusions and provide some analysis, or at least discussion, of how the physics of the system changes as the size of the oscillator network increases.

Our response:

We agree that this statement seems overarching in the previous version of the paper. This statement is in fact underpinned by a combination of the following theoretical considerations:

• Equation (1) is a very general description of an oscillating system, of which time crystals are a subset. Coupling between resonators, nonlinearity, and a parametric pump are ingredients that can be used to effectively model a wide array of physical systems.

• Generically, realistic systems are subject to decoherence, either through internal couplings (as described at the top of page 3 of our paper) or through finite coupling to an environment. In the presence of decoherence, well-defined quantum superpositions (cat states) are lost and the long-time solutions, which are relevant for criterion (ii), can be separated into stationary points and fluctuations around these points, as we discuss in section IV.

• These stationary points are the hot spots seen in the associated probability distribution functions of the semiclassical coherent states, see Eq. (7). These approximate the solutions of a classical system and bear no non-classical character.

• The fluctuation of a system are dealt with after Eq. (8). These fluctuations are characterized by novel resonance frequencies and bandwidths appearing in the power spectral density. As we demonstrate experimentally, all of these features arise in an identical fashion in classical systems as well.

In summary, these considerations, borne largely out of a general theory model for $N$ resonators, led us to conjecture that the classical character of time-crystalline phases is a general property and not a consequence of the specific model we study.

With regards to the case of large N in these systems, the principal features persist, such as N normal modes leading to N instability lobes and a complex solution spacing scaling with N. Consequently, we expect that there will still be a parameter regime where our mean-field analysis holds and time crystal physics can be safely explored.

We would like to emphasize that our discussion includes the general case of $N$ oscillators, though the experimental demonstration was confined to $N=2$. We are happy to follow the referee's recommendation to formulate our conclusion in a less provocative way.

Changes:

We have replaced the relevant text by: ``We conjecture that there is no fundamental distinction between dissipative classical and quantum time-crystalline phases in this type of systems. Furthermore, we anticipate that this conclusion also holds for closed system in the prethermal regime, where time-translation symmetry can indeed be broken.''

Report 2

We thank the referee for their careful reading of our manuscript and appreciate the feedback and questions. Below please find our reply to the points raised in the report.

1. The referee writes:

The consequences of the existence of symmetries in classical and quantum systems are very different, but the reader has the impression that they are the same. Let me illustrate this problem with a simple example of a particle in the symmetrical double-well potential. Quantum eigenstates are superpositions of states located in both the wells, because the symmetry of the double-well potential imposes such a requirement in quantum mechanics. In the classical case, it is not a problem for a particle having a well-defined energy to be located in one of the wells. This fact does not surprise anyone, and to call it a spontaneous symmetry breaking in a similar sense to the quantum case would be misleading.

Our response:

We thank the Referee for this question. To clarify what we mean by spontaneous symmetry breaking, let us consider the aforementioned double-well problem. Regardless of the nature of the underlying degrees of freedom used to describe the Hamiltonian of the system (classical or quantum), a double-well potential will exhibit a symmetry between the two wells. In a quantum system, the symmetry manifests in the possibility to have quantum eigenstates that are a superposition of the particle residing in both wells. A spontaneous symmetry breaking occurs when the particle breaks the potential symmetry and collapses into one of the two wells. As stated by the referee, there can be no spontaneous symmetry breaking in a closed quantum system. The breaking of symmetry, i.e., a collapse to one of the minima, is the result of coupling the quantum particle to a dissipative environment, as was thoroughly discussed in the context of the Caldeira-Leggett model and spin-boson problems, see Annals of Physics 149, 374 (1983) and Rev. Mod. Phys. 59, 1 (1987).

It was shown that at short timescales, one can indeed see quantum coherence of the particle that allows for superposition; at intermediate timescales, the particle collapses and resides in one well in the same way a classical particle would; and at long timescales activation occurs and the particle visits the two wells stochastically with equal probability (restoring the symmetry on average). It is in this sense that we discuss the prethermal time crystal regime, wherein we find that in the semiclassical limit, the system clearly manifests the physics of TTSB, and there is no fundamental difference between the classical and quantum system.

Changes:

We now make this point clearer in the second paragraph on page 3 by adding a short version of the discussion above.

2. The referee writes:

If we consider periodically driven systems, we are dealing with discrete translational symmetry in time, but (as in the above examples) its consequences in quantum and classical systems are different. A single classical particle can follow a periodic trajectory with a period, for example, twice as long as that of a periodic drive, while quantum Floquet states must evolve with the same period. The fact that spontaneous breaking of discrete time translation symmetry occurs in quantum many-body systems (in the thermodynamic limit) is a much less trivial phenomenon than the aforementioned classical periodic trajectory.

Our response:

As shown in Phys. Rev. A 56, 4045 (1997), driven quantum oscillators are special as they have a dense quasi-energy spectra. The presence of multiple weak avoided crossings impact the convergence of any Floquet perturbative or Magnus-like expansion schemes. The presence of such dense energies makes it possible for states to manifest TTSB in states that have a higher period than that of the drive, see Phys. Rev. A 96, 052124 (2017). In this sense, as far as oscillator-based networks are concerned, one can confidently link the TTSB physics of the semi-classical regime to that of the corresponding classical system.

An alternate way of viewing this is through the prism of wave-mixing processes leading to subharmonic responses (period-doubling bifurcations). These processes are rather generic in quantum optics, circuit QED and optomechanics, regardless of the notation used to describe these systems. In line with the fact that phase transitions occur at points where maximal fluctuations appear, and where a classical description suffices, the quantum Floquet expansion must have a good classical limit that captures the same symmetry broken physics.

3. The referee writes:

Another problem that is not clearly presented in the manuscript is the difference between time crystals in closed and dissipative systems. While in the latter it is much easier to deal with the problem of pumping energy as a result of a periodic drive, because the system can release energy to the environment, in the case of the former, breaking ergodicity is highly non-trivial. A generic closed many-body system that is periodically driven should heat up to infinite temperature (in the sense of the eigenstate thermalization hypothesis). Discrete time crystals seem to break this hypothesis. In the current version of the article, the reader may get the impression that there is no significant difference between dissipative time crystals and closed-system time crystals, which is not true.

Our response:

The referee points out that we have insufficiently explained the distinction between closed and dissipative time crystals. Indeed, one of our main points was to elucidate that there is no fundamental distinction between the period-doubling mechanisms in the two cases.

To see this, first consider the dissipative case: the system is driven by the parametric drive and undergoes a period-doubling bifurcation (subharmonic response) due to the interplay between the drive and nonlinearity in the system (many-body interactions leading to wave-mixing processes). Importantly, the relevant modes that gain from the drive become strongly detuned from the drive as they grow in amplitude due to the inherent nonlinearities. This effectively decouples the mode from the drive, thereby limiting the amount of energy pumped into the system. Any excess heat is moved to the bath via dissipation. As such, the system dwells for some time in a subharmonic stationary attractor. As we show in the manuscript, the fluctuations in the system activate stochastic dynamics between the attractors, leading to an effective thermalization in the long-time limit, where the different time symmetry-broken states are sampled probablistically.

In the closed and macroscopic system: we can split the system into modes that react resonantly to the drive and modes that act as a nonresonant background. The resonant modes can undergo a period-doubling bifurcation (subharmonic response) due to the interplay between the drive and nonlinearity in the system (many-body interactions leading to wave-mixing processes). As before, these modes become detuned with respect to the drive and decouple from it due to the nonlinearity, thus limiting the amount of pumped energy. Any excess heat is then dispersed into the nonresonant background, leading at long times to infinite temperatures (ETH). Hence, here too the time symmetry-broken case appears only on prethermal time scales.

We would like to emphasize again that in bosonic nonlinear systems, the energy absorption from the drive is limited by the nonlinearity of the individual oscillators. The amount of coupling to an environment is therefore not crucial for the steady-state amplitude and there is no reason to expect the temperature to grow infinitely. This is still true for normal modes formed by a large number of oscillators coupled within a closed system as we show in section II. For instance, a large mechanical membrane formed by many atoms forms normal vibration modes. When these modes are parametrically driven, their amplitude is only affected by the damping in a minor fashion (by pushing the driving thresholds). We have now added a short clarification of these points in our manuscript.

Changes:

Following the point raised by the referee, we further clarify the role of the nonlinearity to limit the energy pumped into driven systems in the paragraph after Eq. (4).