Topological dynamics of adiabatic cat states

Dear editor,

We would like to thank you for your valuable comments and suggestions concerning the improvements on our manuscript that we have decided to resubmit to SciPost Physics.
We followed your suggestion of a major revision of our manuscript to clarify the main points and
precise their relation to earlier work.

The first outcome of our work is the notion of topological coupling between quantum degrees of freedom, a qubit and two bosonic modes. Our work is the first one to treat within a quantum description both driving modes and the qubit, contrary to Refs. 23,25 which rely on Floquet theory and therefore treat one drive as a classical parameter of the Hamiltonian.
Our full quantum treatment enables a quantitative description of (i) the entanglement between the qubit and the two quantum modes, (ii) the role of the quantum metric and of topological constraint on the nature of the adiabatic states, (iii) the time evolution of the quantum fluctuations of the modes' photon numbers and its relation with Bloch oscillations and Bloch breathing. None of these important properties of the dynamics of the model were discussed previously.
On the technical side, we develop an adiabatic expansion valid to all orders in the adiabatic perturbative parameter, which is original. This reveals the role of a "dressed" Berry curvature in the dynamics instead of the bare Berry curvature. This also enables us to show that (iv) the adiabatic states are not naturally prepared experimentally but generically evolve into
cat states.

We believe that our discoveries meet the acceptance criteria 2 of SciPost Physics. Indeed, the starting point of our work consists in opening a new pathway in the study of topological pumping by considering quantized modes coupled to a qubit instead of a classically driven qubit. In doing so we rely on a separation between fast and slow quantum degrees of freedom instead of a Floquet formalism.
Our change of perspective enables us to demonstrate several fundamental properties of the dynamics of this coupled system. In particular, we unveil topological constraints on the entanglement between the qubit and the modes. This opens a new pathway on the relation between topology and entanglement.

Besides, in the context of Thouless pumps, this opens a new pathway in the understanding of the breakdown of quantization of the pumped charge due to the splitting in a cat state.
For instance one can build on our results to propose protocols of preparation of the adiabatic states which leads to a quantized pumping. We now present these perspectives in the conclusion.


We thank you for your kind consideration.
With our respectful regards,
Jacquelin Luneau,
Benoît Douçot,
and David Carpentier.

Every section has been significantly modified to highlight their main original results. We changed the section 2 to answer valuable questions of the referees concerning how our model of quantum rotor can describe quantum optical system, and what is the physical interpretation of the operators n1 and n2. To clarify these aspects, we added the section 2.1.1 in which we define the notion of topological coupling between two bosonic modes and a two-level system. Doing so, we consider usual types of quantum-optical coupling between a two-level system and the quadratures bosonic modes. The operators n1 and n2 are then naturally the number of quanta in each harmonic oscillator. We added the figure 1 to illustrate this model and the regime of topological coupling, which has never been considered before. We then explain in details in section 2.1.2 that our model of quantum rotors accurately describes the dynamics of quantum states in which the average number of quanta is large compared to their quantum fluctuations. Furthermore, we added a paragraph at the end of Sec. 2.1 establishing the dictionary between our model and a Rice-Mele model with quantum drive. Thus, our results also describe the effect of the quantum nature of a drive on a Thouless pump, which, to our knowledge, has not been discussed before. We added paragraphs translating our results in this context of quantum Thouless pumps at the end of each main section of the manuscript.

In Sec. 3.1, we moved most the technical details in appendix A to highlight the originality of our approach concerning the adiabatic theory: we focus on the nature of the states governed by the effective adiabatic equations of motion, called the adiabatic states. This enables us to show that they are not naturally prepared experimentally, but generically any initial state decomposes into a pair of those, leading to a splitting into a cat state at a topologically quantized rate. We simplified Fig. 4 (former Fig. 3) and Sec. 3.2 on the weight of the cat components to stress the main point: for small quantum fluctuations of the driving phase, this weight is governed by the quantum metric, and the weight of both components are comparable for an initial Fock state due to the topological nature of the coupling. In particular, we moved the secondary results of section 3.3.2 on the preparation of symmetric cat states in appendix E.

In Sec. 4.1, we removed technical derivations and removed the former Fig. 4(d,c) to stress its main results. The entanglement between the qubit and the modes is governed by the geometry of their coupling and enhanced by its topological nature. For small quantum fluctuations of the phase, the entanglement is characterized by the quantum metric. For large quantum fluctuation of the phase (quasi-Fock state), the topological nature of the coupling imposes a strong entanglement.

In Sec. 4.2, we significantly simplified the Fig. 6 (former Fig. 5) to highlight our main results on the quantum fluctuations and temporal evolution of the number of quanta. By increasing the quantum fluctuation of the phases of the mode, we go from a regime of Bloch oscillations to a regime of Bloch breathing. In particular, the quantum fluctuations of the phase stabilizes the topological pumping by reducing the amplitude of the time variation of the center of wavepacket around the topologically quantized drift. In particular, we moved the former subsidiary section 4.2.2 on quasi-periods in appendix B4.

All the modified text appears in blue in the new manuscript.

- We rewrote the abstract. We now precise that our system is the quantum analog of topological pumping. We highlight our results on the characterization of the entanglement between the qubit and the modes by the quantum metric, and on the enhancement of entanglement by the topological nature of the coupling.

- We added a paragraph in the introduction to stress that our definition of cat states does not require an equal superposition of two adiabatic components. We explain that the topological separation in an adiabatic cat states at quantized rate is topologically robust and occurs for any initial state, in contrast with the quantization of pumping which is known to require a well-prepared initial state.

- We added a paragraph in the introduction to highlight our result on the role of the quantum Fubini-Study metric in the entanglement between the qubit and the modes, and to stress that a topological coupling necessarily induces a strong entanglement between a pump and its driving modes.

- We added a paragraph in the introduction detailing the relation of our results with previous works.

- We added the section 2.1.1 to define a notion of topological coupling between two harmonic oscillators and a qubit, in order to clarify the physical meaning of the number of quanta as suggested by Referee 1, and to recover more common types of light-matter interaction as suggested by Referee 2.

- We added the figure 1 to illustrate the model and the regime of topological coupling between two quantum harmonic oscillators and a two-level system.

- We added a paragraph in Sec. 2.1.2 to explain under which conditions our model of quantum rotors is an accurate description of the model of quantum harmonic oscillators, as suggested by Referee 2.

- In Sec. 2.1.2, we merged in one paragraph the discussions of the two possible representations of the state of the modes (the phase and number representation) and the analogy with a particle on a lattice. We explain that the evolution in number representation is solely due to the coupling of the modes to the qubit.

- We moved the numerical values of the Hamiltonian parameters from Sec. 2.2 to Sec. 2.1.2, to provide them after the introduction of the model.

- In Sec. 2.1.2, we added a paragraph detailing the equivalence between our model and a Rice-Mele model of Thouless pump with quantum drive, to stress that our results also describe the effect of the quantum nature of a drive on a Thouless pump.

- We removed the last paragraph of Sec. 2 detailing the organization of the remaining of the article, to avoid redundancies with the end of the introduction.

- At the beginning of Sec. 3, we added a paragraph to stress the originality of our approach concerning the adiabatic theory: we focus on the nature of the states governed by the effective adiabatic equations of motion, called the adiabatic states. This enables us to show that they are not naturally prepared experimentally, but generically any initial state decomposes into a pair of those, leading to a splitting into a cat state at a topologically quantized rate.

- In Sec. 3.1, we now provide only the physical meaning of the adiabatic states and moved the technical details on the definition of the adiabatic projector in appendix A.

- In Sec. 3.1, we added a paragraph to discuss the difference between the adiabatic states and the states considered in previous hybrid classical-quantum descriptions of topological pumps, in terms of the quantum fluctuations of the phase and of the perturbative corrections of the eigenstates in the modes' frequencies.

- In Sec. 3.1., we now stress that the curvature of the adiabatic states is a dressed Berry curvature, in the sense of a perturbative correction of the Berry curvature of the eigenstates.

- We simplified section 3.2 to highlight its main result: any initial state evolves into a cat state splitting at a quantized rate. We moved the expression of the pumping rate in appendix B, and we kept a paragraph discussing the role of the dressed Berry curvature instead of the canonical Berry curvature in the dynamics: the dressed Berry curvature modifies the details of the temporal oscillations of the wavepacket, but does not change the average quantized drift.

- We added a paragraph in Sec. 3.2 to discuss the relation between the topological splitting in a cat states and the breakdown of quantization of a topological pump.

- We removed technical details on the qubit's state representation at the beginning of section 3.3 to summarize the main qualitative result this section: the weights of the two cat components are controlled by the quantum metric for small quantum fluctuations of the phase, and are comparable for large quantum fluctuations of the phases of the modes, due to the topological nature of the coupling.

- We moved Sec 3.3.2 on symmetric cat states to appendix E.

- We significantly simplified figure 4 (former Fig. 3) to highlight the main results of section 3.3. We moved the former Fig. 3(d) concerning symmetric cat states in Fig. 8 of appendix E.

- We added a paragraph in the end of Sec. 3.3 to discuss the relations between our quantitative results on the weight of the cat state and previous studies on the breakdown of quantization of topological pumps.

- We changed the title of Sec. 4 from "Characterization of cat components" to "Entanglement and dynamics of cat states".

- We simplified section 4.1 and Fig. 5 to highlight their main result: (i) the entanglement between the modes and the qubit is controlled by the geometry of the adiabatic states (their quantum metric), and (ii) the topological nature of the coupling leads to a strong entanglement. We removed the subfigures (c) and (d) from Fig. 5 (former Fig. 4)

- We removed technical details at the beginning of Sec. 4.1 and keep the main quantitative result Eq. (20): the qubit is in a statistical mixture of the adiabatic states weighted by the phase density of the modes.

- We simplified section 4.1.2 on the entanglement of quasi-Fock states. We moved the details on the phase distributions of each cat component in appendix E.2 to stress the main result of this section: due to the topological nature of the coupling, large quantum fluctuations of the phase of the modes induce a high entanglement between the qubit and the modes.

- At the beginning of Sec. 4.2, we added references on the manifestation of Bloch oscillations in artificial lattices.

- In Sec. 4.2.1, we simplified the notation of the average and spreading of the number of quanta.

- We moved the former section 4.2.2 on quasi-periods to appendix B.3.

- We modified section 4.2.2 to highlight its main result: topological pumping between quantum modes is "more quantized" than topological pumping between classical modes.

- We moved the discussion on Wannier states leading to no temporal fluctuations of pumping in a separate paragraph of Sec. 4.2.2 for clarity.

- We simplified section 4.2.3 by moving the technical expression of the variance of classical trajectories in appendix B.3, and keeping only the discussion of its role in the refocusing of the wavepacket.

- We changed figure 6 (former Fig. 5) on Bloch oscillations and Bloch breathing to clarify the main points. We replaced the former figures 5(a-d) by Fig. 6(a) containing the definition of the quantities studied in the figure (the average and spreading of number of quanta), and by the insets of Fig. 6(b,d) illustrating the breathing of cat components. We removed the former figures 5(e,g,j) to keep only figures 6(b-e) illustrating the main results of the section: by increasing the quantum fluctuation of the phase, we reduce the amplitude of the Bloch oscillations of the number of quanta and increase the amplitude of their Bloch breathing.

- We moved the role of adiabatic projection on refocusing of adiabatic states to a separate paragraph of Sec. 4.2.3 for clarity.

- In the conclusion, we now highlight our results on adiabatic perturbation theory to all orders and on the relation between entanglement and topology.

- In the conclusion, we added a paragraph showing how our results shed light on the effects of the quantum nature of the drive of a Thouless pump.

- We removed the appendix on Gaussian phase states.