Effective theory for matter in non-perturbative cavity QED

1) First, our claim that “the Hamiltonian presented in panel b) describes a broad range of models” may seem pretentious at face value, but it simply refers to the fact that it is a general Hamiltonian in which some bosonic modes are linearly coupled to an unspecified subsystem. It can be considered general for two reasons, because the nature and geometry of the coupling is not fixed and because many microscopic models, either exactly or approximately, can be cast into this form. Naturally, if approximations are involved in reaching the Hamiltonian in Eq. (17), this restricts its applicability to the regime where the approximations are valid. But this consideration is specific to particular models. This is related to the second part of the comment, which rightfully points out that in reaching this Hamiltonian from the microscopic model of a gas of charged particles coupled to a cavity, we made an assumption about the conservation of momentum. However, this approximation is specific to this model. Other models do not require it and still map exactly to the Hamiltonian in Eq. (17), for instance a system of localized emitters, which is also discussed in the main text (although the calculations are in the Appendix A). Another example of a system that exactly maps to the Hamiltonian in Eq. (17) is a system of magnetic spins, such as the one discussed in Sec. VI C. Surely there are other examples outside of the realm of cavity QED.

In summary, we consider the claim to be fair. We do not claim the Hamiltonian in Eq. (17) to be all-encompassing, that would be an exaggeration. But “broad” seems adequate for a Hamiltonian that can describe bosonic, fermonic and spin systems coupled to bosonic modes with an arbitrary geometry of the coupling.

2) This criticism is addressed in the new section of the manuscript, were the theory is compared with other effective theories and the surname non-perturbative is contextualised. We copy here the relevant paragraphs:

“In this regard, it must be noted that our effective Hamiltonian was derived in the $N \to \infty$ limit. As such, it can be regarded as a zeroth order perturbation theory with $1/N$ as the perturbative parameter. The extension to a full-fledged perturbation theory that can be taken to any order in $1/N$ is a work in progress. However, in this context we will use the term perturbative to refer only to parameters that affect the light-matter coupling regime, such as the light-matter coupling strength or the detuning between the cavity frequency and the characteristic energy scale of the matter subsytem. It is in this sense that our theory is non-perturbative. The simplest example of a perturbative theory would be standard perturbation theory on the light-matter coupling. To yield an effective Hamiltonian with perturbation theory, a separation of energy scales (large detuning) is required, with the more energetic sector of the spectrum mediating effective interactions on a low-energy subspace. […] These numerical results imply an important advantage of our effective theory in regards to its applicability to any light-matter coupling regime, hence the surname non-perturbative. They also prove that it is not equivalent to standard perturbation theory.”

1) We respect the criticism, but we would like to argue that new “physical prediction/insight” were already provided in a previous publication where we applied a more barebones version of the theory to a system of magnetic molecules [10.1103/PhysRevLett.127.167201]. With the current manuscript, we wanted to take the opportunity to develop and highlight the theory in its general version and how it works in some models of current/recent interest. Moreover, following the referee’s comments (see Comment 2) we added a new section dedicated to a thorough testing of the validity of the method, including a comparison with other commonly used methods.

2) This is a fair criticism. In the new version of the manuscript we add a new section that tackles this issue. The validity of the effective theory is tested on the Dicke model by highlighting the differences with common techniques based on the adiabatic elimination of fast variables. We also address its applicability to finite size systems. We copy the conclusions here “These numerical results imply an important advantage of our effective theory in regards to its applicability to any light-matter coupling regime, hence the surname non-perturbative. They also prove that it is not equivalent to standard perturbation theory.”

3) When we refer to the Hamiltonian in Eq. (17) as general, we say it to mean that can describe a broad range of models, not that it describes all those models exactly. For instance, it is true that for a gas of charged particles the approximation of momentum conservation has to be applied. This restricts the validity of the Hamiltonian to gases where momentum is conserved. Nevertheless, the Hamiltonian can also describe a system of localized emitters, this time without approximations (See our Sec III B). Furthermore, these emitters can be electric of magnetic dipoles. The Hamiltonian in Eq. (17) also accommodates any cavity-geometry, not no mention that the bosonic subsystem need not originate from the microscopic description of a cavity, it may also describe a system of magnons for example. We believe that some emphasis on generality is justified.

Regarding the particular approximation in Eq. (13), we concede in the text that our analysis of its validity is not complete. But we agree with the referee that some additional clarification on its consequences is required. We have updated the manuscript accordingly. We copy here the new paragraph: “The approximation is exact for a homogeneous gas. Furthermore, for gasses that have a second order phase transition to a non-homogeneous phase, the approximation is valid in the vicinity of the transition [10.1103/PhysRevB.100.121109]. This suggests that the resulting model can be used to predict and locate transitions to non-homogeneous gas phases. It is expected that the effect of the approximation will be quantitative and will skew the calculation of observables in the non-homogeneous phase.”

Finally, as we detailed in our response to comment 2), in the new section of the manuscript we compare the effective theory against standard perturbation theory. They are shown not to be equivalent in all regimes. Although they do coincide in the fast-cavity limit, as the referee rightfully points out.

4) For a magnetic material, in the context of Sec. 5C, standard ferromagnetism arises from the exchange interaction (Pauli’s exclusion principle) whereas photon condensation, or rather its manifestation on the matter subsystem in the form of a cavity-mediated position-dependent all-to-all spin-spin interaction, arises due to the coupling to a sufficiently confined EM field. They are unrelated in their origin. Furthermore, the exchange interaction depends on the overlap of the electron’s wave functions, and thus is rather short-ranged. In contrast, the cavity mediated interaction is long-ranged, for instance for two spins placed at antinodes of the cavity field. For a ferromagnetic material coupled to a cavity, the intrinsic and cavity-mediated magnetic interactions are combined constructively or destructively (depending on the geometry of the coupling to the cavity and the nature of the intrinsic interaction). Now, as we show in Sec. IV C, matter and light observables are related, so it is the case that if a magnetic material coupled to a cavity orders ferromagnetically (be it due to intrinsic or mediated interactions, or their combination) a photonic population appears in the cavity. It is in this sense that photon condensation and ferromagnetism can be linked.

A similar reasoning can be applied to a ferroelectric material, with the caveat that electric coupling is affected by the no-go theorem for homogeneous coupling to the cavity fields.

5) We want to clarify that even in 3D the thermodynamic limit is not trivially free space electromagnetism. This is because the number of particles scales accordingly, filling this infinite space. Intuitively, when we neglect the effect of electromagnetic field on a material in free space, it is because the material fills a negligible volume of this free space. Because the energy of the EM vacuum is spread across all of free space, only a negligible amount couples to the material. When one considers a cavity, it has the effect of confining the EM field such that most of it interacts with the material. Any cavity-mediated effects originate from this confinement. Making the cavity volume infinite does not negate this effect if the material grows accordingly and the filling factor is maintained. Looking at the Hamiltonian, it can be seen that the coupling term is extensive and does not become negligible in this limit. This is the mathematical manifestation of the above argument.

6) The referee is indeed right in their concern about gauge dependencies of the effective theory. It is, in a sense, gauge dependent. However, this does not mean that it cannot be used to make gauge invariant predictions nor does it conflict with our emphasis about gauge invariance during the derivation of the microscopic model. The reason for our insistence on gauge-invariance during the Introduction and the derivation of the microscopic models, up to the Hamiltonian in Eq. (17), stems from the fact that it has been a historically controverted topic in the field. The omission of the $A^2$ term leads to the incorrect prediction of superradiance in models that lack it. The truncation of matter energy levels is also gauge-sensitive. Our emphasis on working with a gauge-invariant microscopic model aims to avoid these pitfalls. Now, because our effective theory traces out the “light” degrees of freedom, it is intrinsically gauge dependent. This is because the separation between “light” and “matter” subsystems is itself gauge dependent. Photons in the Coulomb gauge are excitations of the transverse cavity fields, whereas in the dipole gauge they also incorporate longitudinal fields associated with the polarization field in the matter. This important issue is not downplayed in the manuscript, we devote the third paragraph of the introduction to discuss it. Nazir and Stokes have discussed this concept extensively, see arXiv:2009.10662. Our choice of the Coulomb gauge responds to this issue. In the Coulomb gauge the EM fields are transverse and thus electromagnetic interactions within the material appear as instantaneous Coulomb interactions included in the matter subsystem. This makes sense in this context because these interactions are fully relevant when the material is in free space, in fact, they help explain much of the structure and properties of most materials. By choosing the Coulomb gauge, the effective interactions that appear on the effective action are solely mediated by the transverse fields of the cavity and the intensity is dependent on the degree of confinement of the EM fields. I.e. they reveal the impact of coupling the material to a confined EM field, as opposed to it being in free space. Any observable, consider for instance the full electric field or some correlation function, has its expression in the Coulomb gauge, which may include a combination of light and matter operators. The expectation values of the former can be related to the expectation values of other matter operators using the recipes in Sec. IV C. The expectation value of the resulting matter-only operator can be computed within the effective theory. The key to making gauge-invariant predictions on observables is thus to express them in terms of light and matter operators as prescribed by the Coulomb gauge, as it is in this gauge where the light is traced out.

7) This issue is also addressed in the new section of the manuscript. As we mention there, “our effective Hamiltonian was derived in the $N \to \infty$ limit. As such, it can be regarded as a zeroth order perturbation theory with $1/N$ as the perturbative parameter. The extension to a full-fledged perturbation theory that can be taken to any order in $1/N$ is a work in progress.”. To alleviate the lack of an analytical approach to the N →∞ limit, we perform a numerical study of the finite size scaling of the theory. As we already commented above, we also show that perturbations in $1/N$ are not equivalent to standard perturbation theory on the light matter coupling. Finally, as per our response to comment 6), we believe this is unrelated to gauge issues.

8) The referee is correct in pointing out that these sums over k in the effective theory can lead to UV divergences. This is pragmatically resolved by introducing a frequency cutoff. This is often the case in cavity QED, and it is physically justified by the limitations of the hardware, i.e. real cavities cannot host infinite frequency modes. In the particular case of the Hamiltonians in Eq. 49 or Eq. 76, it does indeed lead to a mass renomarlization. For the case of a 2D electron gas in the long wavelength limit this model and its mass renormalisation is discussed in Ref. 10.1103/PhysRevResearch.4.013012. The cavity field is shown to dress the electrons, increasing their effective mass. Ultimately, the cutoff cannot be arbitrarily large: the renormalised mass is shown to become infinite when the cutoff is raised to the Landau pole, signalling the breakdown of the effective theory.

UV cutoffs are also commonly employed in dissipative models with ohmic spectral functions. In the context of the path integral, what we denote as the kernel $\bar{\bar K}_\kappa(\tau)$ in Eq. (39) is divergent in models with ohmic damping. This divergence is again tamed by introducing a high frequency Drude cutoff to the spectral function [10.1007/3-540-45855-7_1].

9) We have added a paragraph on Sec. IV C (Relation between matter and light observables) to discuss the probing of cavity qed materials and how the observables are related to the relations that we present in this section. We copy it here for convenience:

“These relations may be important for computing measurable outputs of a cavity QED material. For example, if we probe the cavity-matter system through the transmissivity of the former, the transmitted signal is proportional to the dynamical response of the system. This is in turn related to two point correlators of bosonic fields that can be directly computed from relation (46) or analogues”