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Nonequilibrium quasiparticle distribution in superconducting resonators: effect of pair-breaking photons
by Paul B. Fischer, Gianluigi Catelani
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Submission summary
| Authors (as registered SciPost users): | Gianluigi Catelani |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2401.12607v2 (pdf) |
| Date accepted: | 2024-08-15 |
| Date submitted: | 2024-07-04 07:49 |
| Submitted by: | Catelani, Gianluigi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Many superconducting devices rely on the finite gap in the excitation spectrum of a superconductor: thanks to this gap, at temperatures much smaller than the critical one the number of excitations (quasiparticles) that can impact the device's behavior is exponentially small. Nevertheless, experiments at low temperature usually find a finite, non-negligible density of quasiparticles whose origin has been attributed to various non-equilibrium phenomena. Here, we investigate the role of photons with energy exceeding the pair-breaking threshold $2\Delta$ as a possible source for these quasiparticles in superconducting resonators. Modeling the interacting system of quasiparticles, phonons, sub-gap and pair-breaking photons using a kinetic equation approach, we find analytical expressions for the quasiparticles' density and their energy distribution. Applying our theory to measurements of quality factor as function of temperature and for various readout powers, we find they could be explained by assuming a small number of photons above the pair-breaking threshold. We also show that frequency shift data can give evidence of quasiparticle heating.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Dear Editor,
we submit our revised manuscript for your further consideration. We include below detailed responses to the Referees' reports, which contain descriptions of the changes made to the article.
Response to Report 1:
We thank the Referee for carefully reading the manuscript, recommending its acceptance, and suggesting improvements in the presentation. In the revised manuscript, we have added a new Fig.1 that represent the main system setup we investigate and mention more explicitly what distinguish the present work from the previous one, for instance in the captions to Figs. 1 and 2 and at the beginning of Sec.III. More discussions of our results have been added, for example at the end of Sec.III and in Sec.V.
Regarding the three specific points raised in the report, we have addressed them as follows:
1. We have clarified the processes included in the photon collision integral and the origin of the multiple peaks in the text after Eq.(39)
2. We now state clearly that the “second peak” of magnitude 10^{-18} in Fig. 7 (former Fig.6) is not relevant, see text after Eq.(48).
The experimentally relevant order of magnitude for the value of the distribution function near the gap is discussed at the end of Sec.III by relating it to the normalized quasiparticle density x_qp.
The details about the peaks are not relevant to estimate the quality factor, see text in brackets after Eq.(55).
3. We agree with the Referee that the assumption of a single high-frequency mode can be an idealized one in some cases, such as bulk cavities. The experimental data we analyze is for a coplanar waveguide resonator; in 2D chips, one or a few spurious modes could dominate the high-frequency response. Also, if the noise source is indeed the microwave generator, the high frequency photons could be emitted with sufficient rate only at one or a few frequencies. Therefore, the assumption is not unrealistic. In any case, parts of our calculations are straightforward to generalize for multiple modes. We had already pointed this out briefly at the very end of Sec.IV, we have now added mentions of such possible extensions also in the text preceding subsection III.A and that preceding subsection V.A
Response to Report 2:
We thank the Referee for finding our work “serious and useful” and for making several recommendations for improvements. In our view, the manuscript satisfies at least one of the “novelty” criteria to be accepted in SciPost Physics: the issue of the low-temperature saturation of the quality factor in superconducting devices is a long-standing problem, and our results point to the significant impact that a small number of pair-breaking photons can have on this quantity, an effect that was not previously appreciated. In the first version of the article, this was perhaps not made sufficiently clear. In the revised version, we now emphasize that the experimental data of Ref. 18 cannot be explained by standard approaches, such as losses due to two-level systems, while our approach is able to fit the data using a minimal number of additional parameters.
We specify below the changes implemented to address the detailed questions of the Referee. Together with other improvements (for instance, the addition of the new Fig.1), we believe that the revised manuscript is appropriate for publication in SciPost Physics.
Page 1: We have expanded the introduction to clarify the heating mechanism by which low-energy photons lead to the creation of additional quasiparticles. The Referee is correct in saying that this mechanism is not very efficient: while it can eventually lead to a saturation of the quality factor, the predicted value for the latter is much higher than what experimentally measured, as we now state. This was in fact a key finding of our previous work, Ref.17: low-energy photons alone cannot explain the data.
Page 1 (now page 2): The use of a single mode could be justified in some situations – this could be for instance a spurious mode in a planar chip, or noise generated at a specific frequency by control electronics. We now mention how the approach can be extended to multiple modes, see for example the text after Eq.(3), that before Sec. III.A, the end of Sec.IV (already in the first version), and in particular the text before Sec. V.A, where we explain why the assumption does not qualitatively affect our results.
Page 2, Eq.(2): The reason why a single non-pair-breaking photon is considered is now explained in the text after Eq.(3). Throughout the text, we related terms in the collision integrals to the diagrams in Fig.2 (former Fig.1).
Page 2, Eq.(4) and (5) and Page 2, before Eq.(4): These points are addressed in the previous two points.
Page 2, Fig. 1 (now Fig.2): As explained in Sec. III.A, photon assisted recombination is negligible compared to phonon assisted recombination. For added clarity, this is now mentioned in the caption of Fig.2 as well.
Diagram g) leads to a renormalization of the recombination coefficient, we have added reference to this diagram after Eq.(8).
Diagram h) affects the density through the G(T_*/Δ) term and is only important for very large photon numbers. This is now mentioned in the caption of Fig 2 and we refer to diagram h) before Eq.(50).
Page 3, above Eq. (6): The consistency of the f<<1 assumption should in fact be verified once the distribution functions has been calculated, as we now state in the text before Eq.(6). For the regime of low quasiparticle densities we consider, this assumption is in general valid.
When making approximations it is important that quasiparticle number conserving terms remain number conserving; this is the case when neglecting Pauli blocking.
Page 3, Eq.(6): The notation for energy \epsilon is now more clearly defined after Eq.(6): \epsilon=0 correspond to the BCS gap energy E=\Delta. We have also added the definition of St_g^{Phon}. The phonon occupation does not appear because, as mentioned before Eq. (6), the phonon temperature is assumed to be sufficiently low to permit ignoring phonon absorption by quasiparticles; this assumption is discussed in more detail in Sec. III.A.
Page 3: We have improved section two by: adding a new Fig.1; relating the diagrams in Fig.2 (former Fig.1) to the collision integral; clarifying the notation and definitions of various terms.
Page 3, Eq.(12): The function U(E_1,E_2) has a divergence only for E_2 approaching \Delta while is regular for E_1 near \Delta, see the definition in the text after Eq.(3). Since E+\omega_PB > 3\Delta, we approximate U with the value it takes for large second argument.
Page 4-5: We now discuss the difference between \gamma’_* and \gamma_* in the text after Eq.(28)
Page 6, Fig.4 (now Fig. 5): We denote the black line with the label “numeric”
Page 7: We briefly discuss the results of the section and comment on the experimentally relevant values for f at the end of Sec.III.
Page 10: We briefly describe the derivation of Eq.(49) in the text after Eq.(50)
Page 11, Eq.(54-55): The values of Q_{I,ext} are those shown in Fig.10 (former Fig.9), see also text after Eq.(55). Their overall decrease with increasing power rules out an explanation in terms of dielectric losses due to two-level systems; we now make this important point in the introduction as well as in Sec.V.
The fitting in Fig.11 (former Fig.10) is fully based on our theory, without any additional extrinsic mechanisms, and requires fewer parameters than a phenomenological fit. We remark on this in the paragraph after that containing Eq.(56).
The properties of the resonator enter implicitly into the problem via the photon frequency \omega_0 and the coupling constant c^{QP}_{Phot} (for the latter, see Appendix A, Ref.17 and references there).
Page 13, Eq.(59): We briefly discuss the origin of the equation in the text preceding it. The relation to Kramers-Kronig is mentioned in the conclusions.
we submit our revised manuscript for your further consideration. We include below detailed responses to the Referees' reports, which contain descriptions of the changes made to the article.
Response to Report 1:
We thank the Referee for carefully reading the manuscript, recommending its acceptance, and suggesting improvements in the presentation. In the revised manuscript, we have added a new Fig.1 that represent the main system setup we investigate and mention more explicitly what distinguish the present work from the previous one, for instance in the captions to Figs. 1 and 2 and at the beginning of Sec.III. More discussions of our results have been added, for example at the end of Sec.III and in Sec.V.
Regarding the three specific points raised in the report, we have addressed them as follows:
1. We have clarified the processes included in the photon collision integral and the origin of the multiple peaks in the text after Eq.(39)
2. We now state clearly that the “second peak” of magnitude 10^{-18} in Fig. 7 (former Fig.6) is not relevant, see text after Eq.(48).
The experimentally relevant order of magnitude for the value of the distribution function near the gap is discussed at the end of Sec.III by relating it to the normalized quasiparticle density x_qp.
The details about the peaks are not relevant to estimate the quality factor, see text in brackets after Eq.(55).
3. We agree with the Referee that the assumption of a single high-frequency mode can be an idealized one in some cases, such as bulk cavities. The experimental data we analyze is for a coplanar waveguide resonator; in 2D chips, one or a few spurious modes could dominate the high-frequency response. Also, if the noise source is indeed the microwave generator, the high frequency photons could be emitted with sufficient rate only at one or a few frequencies. Therefore, the assumption is not unrealistic. In any case, parts of our calculations are straightforward to generalize for multiple modes. We had already pointed this out briefly at the very end of Sec.IV, we have now added mentions of such possible extensions also in the text preceding subsection III.A and that preceding subsection V.A
Response to Report 2:
We thank the Referee for finding our work “serious and useful” and for making several recommendations for improvements. In our view, the manuscript satisfies at least one of the “novelty” criteria to be accepted in SciPost Physics: the issue of the low-temperature saturation of the quality factor in superconducting devices is a long-standing problem, and our results point to the significant impact that a small number of pair-breaking photons can have on this quantity, an effect that was not previously appreciated. In the first version of the article, this was perhaps not made sufficiently clear. In the revised version, we now emphasize that the experimental data of Ref. 18 cannot be explained by standard approaches, such as losses due to two-level systems, while our approach is able to fit the data using a minimal number of additional parameters.
We specify below the changes implemented to address the detailed questions of the Referee. Together with other improvements (for instance, the addition of the new Fig.1), we believe that the revised manuscript is appropriate for publication in SciPost Physics.
Page 1: We have expanded the introduction to clarify the heating mechanism by which low-energy photons lead to the creation of additional quasiparticles. The Referee is correct in saying that this mechanism is not very efficient: while it can eventually lead to a saturation of the quality factor, the predicted value for the latter is much higher than what experimentally measured, as we now state. This was in fact a key finding of our previous work, Ref.17: low-energy photons alone cannot explain the data.
Page 1 (now page 2): The use of a single mode could be justified in some situations – this could be for instance a spurious mode in a planar chip, or noise generated at a specific frequency by control electronics. We now mention how the approach can be extended to multiple modes, see for example the text after Eq.(3), that before Sec. III.A, the end of Sec.IV (already in the first version), and in particular the text before Sec. V.A, where we explain why the assumption does not qualitatively affect our results.
Page 2, Eq.(2): The reason why a single non-pair-breaking photon is considered is now explained in the text after Eq.(3). Throughout the text, we related terms in the collision integrals to the diagrams in Fig.2 (former Fig.1).
Page 2, Eq.(4) and (5) and Page 2, before Eq.(4): These points are addressed in the previous two points.
Page 2, Fig. 1 (now Fig.2): As explained in Sec. III.A, photon assisted recombination is negligible compared to phonon assisted recombination. For added clarity, this is now mentioned in the caption of Fig.2 as well.
Diagram g) leads to a renormalization of the recombination coefficient, we have added reference to this diagram after Eq.(8).
Diagram h) affects the density through the G(T_*/Δ) term and is only important for very large photon numbers. This is now mentioned in the caption of Fig 2 and we refer to diagram h) before Eq.(50).
Page 3, above Eq. (6): The consistency of the f<<1 assumption should in fact be verified once the distribution functions has been calculated, as we now state in the text before Eq.(6). For the regime of low quasiparticle densities we consider, this assumption is in general valid.
When making approximations it is important that quasiparticle number conserving terms remain number conserving; this is the case when neglecting Pauli blocking.
Page 3, Eq.(6): The notation for energy \epsilon is now more clearly defined after Eq.(6): \epsilon=0 correspond to the BCS gap energy E=\Delta. We have also added the definition of St_g^{Phon}. The phonon occupation does not appear because, as mentioned before Eq. (6), the phonon temperature is assumed to be sufficiently low to permit ignoring phonon absorption by quasiparticles; this assumption is discussed in more detail in Sec. III.A.
Page 3: We have improved section two by: adding a new Fig.1; relating the diagrams in Fig.2 (former Fig.1) to the collision integral; clarifying the notation and definitions of various terms.
Page 3, Eq.(12): The function U(E_1,E_2) has a divergence only for E_2 approaching \Delta while is regular for E_1 near \Delta, see the definition in the text after Eq.(3). Since E+\omega_PB > 3\Delta, we approximate U with the value it takes for large second argument.
Page 4-5: We now discuss the difference between \gamma’_* and \gamma_* in the text after Eq.(28)
Page 6, Fig.4 (now Fig. 5): We denote the black line with the label “numeric”
Page 7: We briefly discuss the results of the section and comment on the experimentally relevant values for f at the end of Sec.III.
Page 10: We briefly describe the derivation of Eq.(49) in the text after Eq.(50)
Page 11, Eq.(54-55): The values of Q_{I,ext} are those shown in Fig.10 (former Fig.9), see also text after Eq.(55). Their overall decrease with increasing power rules out an explanation in terms of dielectric losses due to two-level systems; we now make this important point in the introduction as well as in Sec.V.
The fitting in Fig.11 (former Fig.10) is fully based on our theory, without any additional extrinsic mechanisms, and requires fewer parameters than a phenomenological fit. We remark on this in the paragraph after that containing Eq.(56).
The properties of the resonator enter implicitly into the problem via the photon frequency \omega_0 and the coupling constant c^{QP}_{Phot} (for the latter, see Appendix A, Ref.17 and references there).
Page 13, Eq.(59): We briefly discuss the origin of the equation in the text preceding it. The relation to Kramers-Kronig is mentioned in the conclusions.
Published as SciPost Phys. 17, 070 (2024)