Dissipative cooling induced by pulse perturbations

Dear Editor-in-charge,

We have considered the Referees’ reports concerning our manuscript “Dissipative cooling induced by pulse perturbations”, which we submitted to SciPost for publication. First of all, we thank all three the Referees for their work in reviewing our manuscript. That being said, we note that, out of the three Referees, both Referee 1 and Referee 3, during the first round, recognize the validity of the results contained in our manuscript while contributed report of Referee 2 arises a list of remarks:

Referee 1:

1. “The manuscript convincingly presents an interesting result […].”

2. “The mechanism studied in this work could provide an understanding of laser-induced superconductivity.”

Referee 3:

1. “Interesting original ideas.”

During the second round, the Referee 1 (that, looking at the reports, we assume to be the Referee 3 of the first round) still agree on the validity of the results (“I still think that the actual results might warrant publication”), Referee 2 accepted our responses to his remarks and Referee 3 (ex Referee 1) agrees on publication (“I think the resubmission satisfactorily addresses all previous concerns and the manuscript could be published in SciPost Physics.”). However, while during the first round the Referee 1 assert that “The manuscript is very accessible and well-written.”, during the second round Referee 2 and Referee 1 (ex Referee 3) still arise some observation on the way the results are presented.

Below, we provide a detailed answer to the Referees’ remarks, together with a list of the major modifications we correspondingly made to the paper to meet Referees’ criticisms.

We thank you in advance for the kind attention in our submission.

Best Regards Andrea Nava (on behalf of all the authors)

Answers to: Anonymous Report 1 on 2021-9-27 (Invited Report)

[Q1] the question of referee two why the laser-interaction appears as an interaction between two spins is explained in the response of the referees but not in the manuscript

[A1] We further improved the answer to Referee 2, implementing her/his further observations around the laser-interaction Hamiltonian and adding to the manuscript:

[page 6] “It is worth to stress that, in our model spins and, equivalently, hard core bosons, are just a tool to reproduce a Hilbert space comprising four energy states for each site, two, $\ket{0;i}$ and $\ket{1;i}$, with lower energy and two, $\ket{2;i}$ and $\ket{3;i}$, with higher one. The laser pulse allows transitions between the two sectors, specifically, $0 \leftrightarrow 3$ and $1 \leftrightarrow 2$, which are simply interpreted in terms of hard-core bosons. The key to our mechanism is to tune the laser frequency in resonance with $1 \leftrightarrow 2$, thus depleting the occupation of boson $b_{1,i}$ and increasing that of $b_{2,i}$. In this sense (19) mimics a laser pulse, acting at the level of single particles. In analogy with optical absorption of light we can image that $b_{1,i}$ is even under parity and $b_{2,i}$ odd, so that all transition processes $0 \leftrightarrow 3$ and $1 \leftrightarrow 2$ are dipole active, i.e. couple opposite parity states.”

[Q2] While I still think that the actual results might warrant publication, I feel that the explanations are substantially too scarce to make the manuscript reasonably accessible. […] Fig. 4 for example shows several complicated functions, but there is no explanation of what features in the functions are relevant. […] I don't think that many readers will be able to appreciate the results.

[A2] We thank the Referee for her/his suggestion. We believe that, after the further improvements we made in revising our paper, it is now more accessible and clearer. In the resubmitted version of the manuscript, we extended the explanation and discussion of the results to make them more detailed. All the additions are listed in the “List of major changes” section of the resubmission letter.

Answers to: Anonymous Report 2 on 2021-10-12 (Contributed Report)

[Q1] I still don't think that introducing the notion "hard core bosons" is very helpful, but I leave the choice with the authors. I would just ask the authors to please write a few more details or at least mention that this is merely a formal construction. Just introducing the bosons by writing "In other words, we can write..." is not helpful. Also, please use the bosons consistently; e.g. below Eq. (19) you resort to projectors/ket-bra symbols, that could all be represented by bosonic operators for consistency. The same for the jump operators; those could also be expressed in terms of hard core bosons, right?

[A1] We thank the referee for her/his observation. We agree that it would be better to give some more details on the mathematical aspect of the hard core bosons. For this reason, we added to the manuscript, before “the optical analogy”:

[page 6] “It is worth to stress that, in our model spins and, equivalently, hard core bosons, are just a tool to reproduce a Hilbert space comprising four energy states for each site, two, $\ket{0;i}$ and $\ket{1;i}$, with lower energy and two, $\ket{2;i}$ and $\ket{3;i}$, with higher one. The laser pulse allows transitions between the two sectors, specifically, $0 \leftrightarrow 3$ and $1 \leftrightarrow 2$, which are simply interpreted in terms of hard-core bosons. The key to our mechanism is to tune the laser frequency in resonance with $1 \leftrightarrow 2$, thus depleting the occupation of boson $b_{1,i}$ and increasing that of $b_{2,i}$. In this sense (19) mimics a laser pulse, acting at the level of single particles. In analogy with optical absorption of light we can image that $b_{1,i}$ is even under parity and $b_{2,i}$ odd, so that all transition processes $0 \leftrightarrow 3$ and $1 \leftrightarrow 2$ are dipole active, i.e. couple opposite parity states.”

For what concern the consistency related to the hard core bosonic notation, the projectors/ket-bra symbols below Eq.(19) are already expressed in terms of b_1 and b_2 while the Lindblad operators in Eq.(25) can be easily expressed in terms of b_1 and b_2 resorting to Eq.(16). While we find that it would not be helpful to list all the Lindblad operator, one by one, in terms of the hard core bosons, we agree that it is useful to mention it. For this reason, we added, before “Detailed balance”:

[page 8] “Lindblad operators can be easily expressed in terms of hard core bosons through Eq.(16).”

[Q2] Thanks for the clarification. Please rework the sentence that you added to the manuscript. It is not clear what you mean by "optical analogy" (the reader does not know about our exchange here).

[A2] We thank the referee for the suggestion. In [A1] we also rephrased the paragraph regarding the optical analogy including part of our exchange.

[Q3] Can you please spell out what "compatibility with the mean-field character" precisely means?

[A3] In order to make clear what we mean by “compatibility with mean-field character of the Hamiltonian”, we added the following note in the manuscript:

“After resorting to the mean field approximation (let us remember that in our case it is exact in the thermodynamic limit), the Hamiltonian becomes a function of the density matrix. Indeed, the Hamiltonian parameters (the magnetic field in our case) should be determined by self-consistency conditions like in Eq.(23). It follows that the Hamiltonian eigenvalues becomes a function of the density matrix too. Now, as the set of Lindblad operators is written in terms of the Hamiltonian eigenstates, when a mean field approximation is performed the Lindblad equation becomes a non-linear differential equation with Lindblad operators that are implicitly a function of the density matrix. Furthermore, in principle a Lindblad operator could connect any two states within the Hilbert space of an interacting Hamiltonian but, after the mean field approximation, as the density matrix factorises into the product of single-site density matrices, the Lindblad operators only involves eigenstates at the same site. See Refs.[26, 39, 40] for some examples.”

[39] PHYSICAL REVIEW B 100, 125102 (2019), Andrea Nava and Michele Fabrizio, Lindblad dissipative dynamics in the presence of phase coexistence

[40] PHYSICAL REVIEW B 103, 115139 (2021), Andrea Nava , Marco Rossi , and Domenico Giuliano, Lindblad equation approach to the determination of the optimal working point in nonequilibrium stationary states of an interacting electronic one-dimensional system: Application to the spinless Hubbard chain in the clean and in the weakly disordered limit

[Q4] Thank you, I accept that. But could you please explain in what sense the strength of the coupling to the laser pulse is the "most critical parameter"? Also, rather than writing "we make some not unphysical assumptions", you could just spell out once more what assumptions you are making. And after that you can re-iterate why they are reasonable/physical.

[A4] In order to highlight the discussion about the unphysical assumptions and the strength of the coupling to the laser pulse, we added into the conclusions of the manuscript

“It is worth stressing that, while we make some not unphysical assumptions on the dissipative processes, we also show that the selective cooling persists for a wide range of parameters. Indeed, we just assume that high-energy excitations dissipate faster than low-energy ones, which is a rather generic circumstance. In reality, the most critical parameter here is the strength of the coupling to the ‘laser pulse’, i.e., the dipole matrix element in real materials, or, equivalently, the intensity of the optical absorption that is related to the strength of the transition processes. In our toy model there is just a single transition process that connects the low energy sector, states 0 and 1, with the high energy one, states 2 and 3. Evidently, the cooling is less efficient the smaller the coupling strength. However, regarding K3C60, we recall that in the experiment the laser hits a rather pronounced mid-infrared peak, which we have associated to an exciton that plays the role of the high-energy sector [20], while the role of the low energy sector is there played by particle-hole excitations. The intensity of that peak reflects the substantial strength of the process. It is self-evident that it would be totally useless to pump at an excitation with very small absorption strength. Indeed, the strength of the optical process is a prerequisite of our cooling strategy.”

Answers to: Anonymous Report 3 on 2021-10-13 (Invited Report)

We thank the Referee for her/his appreciation of our work.

[Q1] The expression of the interaction with the laser field in terms of two distinct spins actually makes sense because of the vastly different energy scales being involved. Making this point more explicit in the manuscript might be a good idea, but it is not strictly necessary for the understanding of the paper. […] I also think that the authors comment on Fig. 4 in the main text in an adequate way, especially concerning the influence of r on relaxation time.

[A1] We thank the Referee for her/his suggestion. In the revised version of the manuscript, we extended the discussion at page 6 in order to meet other Referees’ request too and added some more comment about the model and the results here and there in the manuscript (as reported in the list of major changes).

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List of major changes
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[page 6] “It is worth to stress that, in our model spins and, equivalently, hard core bosons, are just a tool to reproduce a Hilbert space comprising four energy states for each site, two, $\ket{0;i}$ and $\ket{1;i}$, with lower energy and two, $\ket{2;i}$ and $\ket{3;i}$, with higher one. The laser pulse allows transitions between the two sectors, specifically, $0 \leftrightarrow 3$ and $1 \leftrightarrow 2$, which are simply interpreted in terms of hard-core bosons. The key to our mechanism is to tune the laser frequency in resonance with $1 \leftrightarrow 2$, thus depleting the occupation of boson $b_{1,i}$ and increasing that of $b_{2,i}$. In this sense \eqn{eq:envelope} mimics a laser pulse, acting at the level of single particles. In analogy with optical absorption of light we can image that $b_{1,i}$ is even under parity and $b_{2,i}$ odd, so that all transition processes $0 \leftrightarrow 3$ and $1 \leftrightarrow 2$ are dipole active, i.e. couple opposite parity states.”

[page 8] “Lindblad operators can be easily expressed in terms of hard core bosons through Eq.(16).”

[page 9] “In Fig. 4 we show the results of the numerical integration of Eq. (24) at temperature T = 1.5 Tc, and for increasing r, see Eq. (27), from r = 1, blue line, to r = 640, purple line. The laser pulse parameters, see Eq. (19), are E0 = 0.2 and τ = 1000. As long as the laser pulse is on, the order parameter time evolution is similar to the non dissipative case shown in Fig. 2, i.e. it starts from $m_{x,1}=0$, corresponding to the disordered system, and evolves toward one of the two Z2-equivalent symmetry ordered phases. At the end of the laser pulse, while in the non dissipative case the system remains trapped in the ordered phase with $m_{x,1}\neq0$, in the dissipative case the systems tends to return in the disordered one after a time that depends on the ratio r. We note that when low-energy excitations dissipate as fast as high-energy ones, blue line at r = 1, the system quickly relax to thermal equilibrium, $m_{x,1}=0$, without showing any transient cooling. The latter appears only upon increasing r, and lasts longer the larger r is. To quantify how long […]”

[page 13] “Again, the result shows an interesting qualitative agreement with the experimental observation reported in panel a) of Fig.5 of Ref. [14] where, indeed, ’the photoresistivity was mostly independent of the pump-pulse duration and depended only on the total energy of the excitation pulse’. Indeed, as observed in Ref.[14], this is a rather interesting aspect as the fact that the amplitude of the effect only depends on the laser pulse fluence is a strong evidence against the early proposed approaches based on the non-linear phonon mechanism where the system response is expected to depend on the laser pulse peak amplitude [12] while it looks to be consistent with the selective cooling mechanism.”

[page 13] “It is worth stressing that, while we make some not unphysical assumptions on the dissipative processes, we also show that the selective cooling persists for a wide range of parameters. Indeed, we just assume that high-energy excitations dissipate faster than low-energy ones, which is a rather generic circumstance. In reality, the most critical parameter here is the strength of the coupling to the ‘laser pulse’, i.e., the dipole matrix element in real materials, or, equivalently, the intensity of the optical absorption that is related to the strength of the transition processes. In our toy model there is just a single transition process that connects the low energy sector, states 0 and 1, with the high energy one, states 2 and 3. Evidently, the cooling is less efficient the smaller the coupling strength. However, regarding K3C60, we recall that in the experiment the laser hits a rather pronounced mid-infrared peak, which we have associated to an exciton that plays the role of the high-energy sector [20], while the role of the low energy sector is there played by particle-hole excitations. The intensity of that peak reflects the substantial strength of the process. It is self-evident that it would be totally useless to pump at an excitation with very small absorption strength. Indeed, the strength of the optical process is a prerequisite of our cooling strategy.”

[note] “After resorting to the mean field approximation (let us remember that in our case it is exact in the thermodynamic limit), the Hamiltonian becomes a function of the density matrix. Indeed, the Hamiltonian parameters (the magnetic field in our case) should be determined by self-consistency conditions like in Eq.(23). It follows that the Hamiltonian eigenvalues becomes a function of the density matrix too. Now, as the set of Lindblad operators is written in terms of the Hamiltonian eigenstates, when a mean field approximation is performed the Lindblad equation becomes a non-linear differential equation with Lindblad operators that are implicitly a function of the density matrix. Furthermore, in principle a Lindblad operator could connect any two states within the Hilbert space of an interacting Hamiltonian but, after the mean field approximation, as the density matrix factorises into the product of single-site density matrices, the Lindblad operators only involves eigenstates at the same site. See Refs.[26, 39, 40] for some examples.”

We added the following references:

[39] PHYSICAL REVIEW B 100, 125102 (2019), Andrea Nava and Michele Fabrizio, Lindblad dissipative dynamics in the presence of phase coexistence

[40] PHYSICAL REVIEW B 103, 115139 (2021), Andrea Nava, Marco Rossi , and Domenico Giuliano, Lindblad equation approach to the determination of the optimal working point in nonequilibrium stationary states of an interacting electronic one-dimensional system: Application to the spinless Hubbard chain in the clean and in the weakly disordered limit