Exact solution of polaritonic systems with arbitrary light and matter frequency-dependent losses

This figure displays the light |K_±,k|² [(a) and (b)] and matter |J_±,k|² [(d) and (e)] components of the two polaritonic branches, as wells as their sums |K_k|² (c) and |J_k|² (f), when the gauge is fixed by s_x,k,− = 0. The field spectra are plotted as functions of the bare cavity frequency ω_k (red dashed line), while the resonant matter frequency ${\tilde{ω}}_{x}$ is fixed and used as a unit of frequency (black dotted line). Other parameters are g = 0.3ω_x and γ_P = γ_M = 0.05ω_x. The field spectra are here normalized to the maximum value for all the plots in the same row. The calculated polariton modes in the lossless case ω_−,k and ω_+,k are marked by a dotted-dashed blue lines. Although we maintain the notation K_±,k and J_±,k, it is clear that it is no longer possible to isolate the contributions of the two polariton branches.

FIG. 2.

The panels display the effects of the interplay between light and matter losses on the analytically derived dressed photonic |K_k(ω)|² [(a)–(c)] and matter |J_k(ω)|² [(b)–(d)] fields. In all the plot, g = 0.3ω_x, while the losses rates are γ_M = γ_P = 0.4ω_x in (a) and (d), γ_P = 0.05ω_x and γ_M = 0.2ω_x in (b) and (e), and γ_P = 0.2 and γ_M = 0.05ω_x in (c) and (f). The field spectra are normalized to the maximum value for all the plots in the same row. All the other parameters remain as in Fig. 1.

FIG. 3.

The panels display the analytical dressed photonic |K_k(ω)|² [(a)–(c)] and matter |J_k(ω)|² [(b)–(d)] fields varying the light–matter coupling strength g. In all the plot, γ_M = γ_P = 0.05ω_x, while the coupling is g = 0.01ω_x in (a) and (d), g = 0.1ω_x in (b) and (e), and g = 0.3ω_x in (c) and (f). The field spectra are here normalized to the maximum value for all the plots in the same row. All the other parameters remain as in Fig. 1.

FIG. 4.

The panels display the field functions |K_k(ω)|² [(a)–(c)] and |J_k(ω)|² [(b)–(d)], with the presence of a rectangular absorption band centered at frequency ω_c = 2 and of width Δ. The green dotted lines mark the band boundaries. In all the plots, κ = 0.05ω_x; the reservoir losses rates are γ_M = γ_P = 0.05ω_x, and Δ = ω_x [(a) and (d)], Δ = 0.6ω_x [(b) and (e)], and Δ = 0.2ω_x [(c) and (f)]. The field spectra are normalized to the maximum value for all the plots in the same row. All the other features remain as in Fig. 1.

FIG. 5.

Coupled photonic $K_{k}^{2}$ (a) and matter $J_{k}^{2}$ (b) fields as a function of the bare cavity frequency ω_k, calculated starting from a Coulomb representation Hamiltonian. The light–matter coupling is g = 0.3ω_x, while the loss rates are γ_P = γ_M = 0.05ω_x. All the other parameters remain as in Fig. 1.

VI. DIAGONALIZATION WITH COLORED RESERVOIRS

The theory we developed allows us to model polaritonic systems with arbitrary colored reservoirs, including the scientifically and technologically relevant case of a continuum absorption band, a case object of multiple theoretical^22–26 and experimental^27–30 studies. Focusing for the sake of definiteness on a compactly supported reservoir interacting with the photonic component of the excitation (an absorption band), we can include it in the theory by choosing frequency-dependent coupling functions V_k(ω) with support in the chosen frequency range. Exactly the same procedure would couple to the matter component by using Q(ω) instead.

The presence of an absorption band will generally not substitute other loss channels influencing the excitations lifetime and the necessity of keeping both can cause some formal problem given that, as we saw before, the modeling of a Lorentzian resonance requires us to renormalize an otherwise infinite resonant frequency. In this section, we will provide the recipe to add an arbitrary colored reservoir on the top of the Lorentzian one. The microscopic reservoir modes leading to the Lorentzian line shape of the uncoupled photonic resonance are a priori completely uncorrelated from those leading to an absorption band. They will in many cases have even different physical origins, e.g., phonon interaction for the former and interaction with electronic bands for the latter. As better explained in Appendix A, this means their effects sum incoherently and we can thus write the full interaction of the photonic component with its environment using the coupling function,

| V_{k} (ω) |^{2} = | V_{k}^{0} (ω) |^{2} + | V_{k}^{1} (ω) |^{2},

(54)

where $V_{k}^{0} (ω)$ and $V_{k}^{1} (ω)$ model, respectively, the interaction with the photonic reservoir and the absorption band, defined by a normalized density F(ω) with F(ω < 0) = 0, $\int_{0}^{\infty} d ω F (ω) = 1$ ⁠, and a dimensionless coupling intensity κ. The two coupling functions take the form

\begin{gathered} | V_{k}^{0} (ω) |^{2} = \frac{ω {\bar{ω}}_{k}}{q_{P} + ω_{P}} Θ (ω_{P} - ω), \\ | V_{k}^{1} (ω) |^{2} = \frac{ω {\bar{ω}}_{k} q_{P}}{q_{P} + ω_{P}} \frac{κ}{1 + κ} F (ω), \end{gathered}

(55)

where $| V_{k}^{0} (ω) |^{2}$ ⁠, which will lead to the Lorentzian broadening as in the previous case, has the same form as in Eq. (50), and the other term is chosen in order to provide the required absorption band after the renormalization.

The dressed frequency of the photonic mode is now renormalized by both terms. It is practical to write the impact of the Lorentzian broadening as a renormalization over the frequency of the photonic resonance ${\tilde{ω}}_{k}$ dressed only by the absorption band,

{\bar{ω}}_{k}^{2} = {\tilde{ω}}_{k}^{2} + \int_{0}^{\infty} d ω \frac{| V_{k}^{0} (ω) |^{2} {\bar{ω}}_{k}}{ω} = {\tilde{ω}}_{k}^{2} \frac{q_{P} + ω_{P}}{q_{P}},

(56)

and define the latter as

{\tilde{ω}}_{k}^{2} = ω_{k}^{2} + \int_{0}^{\infty} d ω \frac{| V_{k}^{1} (ω) |^{2} {\bar{ω}}_{k}}{ω} = ω_{k}^{2} (1 + κ) .

(57)

We then follow the same diagonalization procedure as in Sec. III, with function χ_k(ω) now taking the form

χ_{k} (ω) = 1 - \frac{1}{2 {\bar{ω}}_{k}} \int_{- \infty}^{\infty} d ω^{'} \frac{V_{k}^{0} (ω^{'})}{ω^{'} - ω + i 0^{+}} - \frac{1}{2 {\bar{ω}}_{k}} \int_{- \infty}^{\infty} d ω^{'} \frac{V_{k}^{1} (ω^{'})}{ω^{'} - ω + i 0^{+}},

(58)

where $V_{k}^{0} (ω)$ and $V_{k}^{1} (ω)$ are the odd analytical extensions in the negative frequency range of $| V_{k}^{0} (ω) |^{2}$ and $| V_{k}^{1} (ω) |^{2}$ ⁠, respectively. The squared function |ζ_k(ω)|² can be written as in Eq. (30),

| ζ_{k} (ω) |^{2} = \frac{| V_{k}^{0} (ω) |^{2} ω^{2} {\bar{ω}}_{k}}{| ω^{2} - {\bar{ω}}_{k}^{2} χ_{k} (ω) |^{2}} + \frac{| V_{k}^{1} (ω) |^{2} ω^{2} {\bar{ω}}_{k}}{| ω^{2} - {\bar{ω}}_{k}^{2} χ_{k} (ω) |^{2}},

(59)

leading to, after some algebra, and after letting ω_P → ∞,

| ζ_{k} (ω) |^{2} = \frac{ω^{3}}{{[ω^{2} - Ω_{k}^{2} (ω)]}^{2} + ω^{2} Γ {(ω)}^{2}} [\frac{2 γ_{P}}{π} + κ ω_{k}^{2} F (ω)]

(60)

with effective central frequency and effective losses

\begin{gathered} Ω_{k}^{2} (ω) = ω_{k}^{2} [1 + κ (1 - \frac{1}{2} P \int_{- \infty}^{\infty} \frac{ω^{'} F (| ω^{'} |)}{ω^{'} - ω} d ω^{'})], \\ Γ (ω) = γ_{P} + \frac{π κ ω_{k}^{2} F (ω)}{2}, \end{gathered}

(61)

where the linewidth of the Lorentzian losses is now defined as a function of the frequency dressed by the absorption band $γ_{P} = \frac{π {\tilde{ω}}_{k}^{2}}{2 q_{P}}$ ⁠. Note that in the low-frequency regime ω → 0 from Eq. (61), we have Ω_k ≈ ω_k, showing that the presence of the absorption band does not change the background permittivity.

From Eq. (60), we see that the photonic losses increase in the frequency region in which F(ω) ≠ 0, an effect already observed in Ref. 30, and it also leads to a resonance effect in the central frequency. This can be understood in light of recent studies on strong coupling with the continuum,^24,26 and we expect it to model the possibility of the photonic resonance becoming strongly coupled with the absorption band. From the renormalized expressions in Eq. (61), the integral function W_k(ω) in Eq. (36) can be simply evaluated numerically.

In Fig. 4, we plot the example results obtained using a sharp absorption band of center frequency ω_c and width Δ,

F (ω) = \frac{1}{Δ} Θ (\frac{Δ}{2} - | ω - ω_{c} |) .

(62)

The boundaries of the band are marked by the horizontal green dashed lines. We recognize the effects expected from our analytical results: The band acts as a localized absorber for the photonic component of the polariton, and eventually, the polaritonic mode gets strongly coupled to the band, an effect better visible when the band width becomes comparable with the intrinsic photonic linewidth.

VII. CONCLUSIONS

In this article, we exactly solved the polaritonic problem with a quantum formalism in the case of arbitrary dissipative couplings for both the bare photonic and matter excitations. In order to do this, we discussed the extension of Fano and HB theories to the case of multiple discrete levels coupled to multiple continua, showing how a gauge indeterminacy emerges. While a previous approach to this problem had been to perform an arbitrary gauge choice, here we analytically calculated the gauge-invariant observables. We thus demonstrated both the self-consistency of our theory and provided an analytical, albeit cumbersome formula allowing to exactly calculate the resonance line shape for strongly coupled resonances interacting with reservoirs of arbitrary spectral shapes. We then showed how colored features can be practically added to a homogeneous resonance linewidth. Note that while our approach is based on a purely bosonic spectrum of the material resonance, generally correct for plasmonic and phononic systems, saturation finite size effects can a priori be included in the theory as higher order terms.⁴² The same is true for generic terms present in the PZW Hamiltonian nonlinear in the light and matter fields, which can be written using Eq. (43) as nonlinear polaritonic interactions.

We hope these results will be of use in the subfields of polaritonic in which losses have an important impact. This is true, for example, in plasmonic systems, characterized by important Ohmic losses, nonlocal systems where photonic excitations couple to a continuum of propagative modes, and in systems in which the extremely large coupling between light and matter pushes the polaritonic resonances into other spectral features.^26,28 In order to facilitate the use of our results by the community, we uploaded on GitHub a Matlab code implementing them.⁴³

ACKNOWLEDGMENTS

S.D.L. is a Royal Society Research Fellow and was partly funded by the Philip Leverhulme Prize of the Leverhulme Trust. The authors acknowledge funding from the RGF\EA\181001 grant from the Royal Society and the Leverhulme Trust under Grant No. RPG-2019-174.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no Conflict of Interest to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request. The program used to generate all the data is available online in Ref. 43.

APPENDIX A: THE FANO DIAGONALIZATION AND ITS EXTENSION

In this appendix, we briefly discuss the basic idea behind the Fano diagonalization and its extensions to the case of many continua or many discrete levels. Following Ref. 2, we consider the Hamiltonian

\begin{align} H = & ω a^{†} a + \int d ω^{'} ω^{'} b^{†} (ω^{'}) b (ω^{'}) + \int d ω^{'} g (ω^{'}) \\ \times [b^{†} (ω^{'}) a + a^{†} b (ω^{'})], \end{align}

(A1)

where in the original paper a and b(ω) in Eq. (A1) are normalized Hilbert space vectors, but in the present context, they can as well be interpreted as second-quantized annihilation operators. Under the assumption that all the coupled eigenvalues fall inside the initial continuum range, Fano showed how the system can be exactly diagonalized in terms of a hybridized continuum,

P (ω) = x (ω) a + \int d ω^{'} y (ω, ω^{'}) b (ω^{'}) .

(A2)

The discrete mode thus gets dressed by a cloud of continuum excitations, translating into a spectral broadening of the resonance. Notice that this set of solution is not necessarily complete, as known from the study of the Friederichs–Lee model.⁴⁴ In the bound-to-continuum strong coupling regime, discrete modes can emerge from the continuum, as theoretically and experimentally demonstrated in the case of two-dimensional electron gases.^24,29

After having completed the diagonalization procedure, Fano passes to consider the case in which there are N discrete levels and one continuum. Such a problem can be reduced to the one treated above by initially performing a partial diagonalization of one discrete level coupled to the continuum, leading to a novel Hamiltonian in the same form as the initial one, but this time with N − 1 discrete levels. Proceeding by iteration, the system can be solved in terms of a single hybridized continuum of the form

P (ω) = \sum_{n = 1}^{N} x_{n} (ω) a_{n} + \int d ω^{'} y (ω, ω^{'}) b (ω^{'}) .

(A3)

Finally, the case of a single discrete state coupled to N continua is treated, described by the Hamiltonian

\begin{align} H & = ω a^{†} a + \sum_{n = 1}^{N} \int d ω^{'} ω^{'} b_{n}^{†} (ω^{'}) b_{n} (ω^{'}) \\ + \sum_{n = 1}^{N} \int d ω^{'} g_{n} (ω^{'}) [b_{n}^{†} (ω^{'}) a + a^{†} b_{n} (ω^{'})] . \end{align}

(A4)

Such a system can be solved by performing the transformation

{\tilde{b}}_{m} (ω) = \sum_{n = 1}^{N} U_{m n} (ω) b_{n} (ω),

(A5)

where U_mn(ω) is a unitary matrix whose first row is given by

U_{1 n} (ω) = \frac{g_{n} (ω)}{{\tilde{g}}_{1} (ω)}

(A6)

with

{\tilde{g}}_{1} (ω) = \sqrt{\sum_{n = 1}^{N} g_{n} {(ω)}^{2}},

(A7)

transforming Eq. (A4) into the Hamiltonian of one discrete level coupled to a single continuum, plus other N − 1 uncoupled continua,

\begin{align} H & = ω a^{†} a + \sum_{n = 1}^{N} \int d ω^{'} ω^{'} {\tilde{b}}_{n}^{†} (ω^{'}) {\tilde{b}}_{n} (ω^{'}) \\ + \int d ω^{'} {\tilde{g}}_{1} (ω^{'}) [{\tilde{b}}_{1}^{†} (ω^{'}) a + a^{†} {\tilde{b}}_{1} (ω^{'})] . \end{align}

(A8)

APPENDIX B: HUTTNER–BARNETT DIAGONALIZATION

In this appendix, we perform the HB diagonalization of the Hamiltonian H_PB in Eq. (11), describing the interaction between the discrete photon mode and the photonic reservoir, modeled as an ensemble of harmonic oscillators indexed by the continuum frequency ω.

We introduce the bosonic operators describing broadened photons A_k(ω),

\begin{align} A_{k} (ω) & = x_{k} (ω) a_{k} + z_{k} (ω) a_{k}^{†} + \int d ω^{'} [y_{k} (ω, ω^{'}) α_{k} (ω^{'})) \\ (+ w_{k} (ω, ω^{'}) α_{k}^{†} (ω^{'})], \end{align}

(B1)

whose coefficients are chosen so that the operators satisfy the eigenequation

ω A_{k} (ω) = [A_{k} (ω), H_{P B}] .

(B2)

This equation leads to the system between the coefficients

x_{k} (ω) (ω - {\bar{ω}}_{k}) = \frac{1}{2} \int_{0}^{\infty} d ω^{'} [y_{k} (ω, ω^{'}) V_{k} (ω^{'}) - w_{k} (ω, ω^{'}) V_{k}^{*} (ω^{'})],

(B3)

z_{k} (ω) (ω + {\bar{ω}}_{k}) = \frac{1}{2} \int_{0}^{\infty} d ω^{'} [y_{k} (ω, ω^{'}) V_{k} (ω^{'}) - w_{k} (ω, ω^{'}) V_{k}^{*} (ω^{'})],

(B4)

y_{k} (ω, ω^{'}) (ω - ω^{'}) = \frac{1}{2} [x_{k} (ω) - z_{k} (ω)] V_{k}^{*} (ω^{'}),

(B5)

w_{k} (ω, ω^{'}) (ω + ω^{'}) = \frac{1}{2} [x_{k} (ω) - z_{k} (ω)] V_{k} (ω^{'}) .

(B6)

This set of equations can be solved to obtain z_k(ω), y_k(ω, ω′), and z_k(ω, ω′) in terms of x_k(ω). By subtracting Eq. (B4) from Eq. (B3), we obtain

z_{k} (ω) = \frac{ω - {\bar{ω}}_{k}}{ω + {\bar{ω}}_{k}} x_{k} (ω),

(B7)

which can be substituted in Eqs. (B5) and (B6) to obtain

\begin{gathered} y_{k} (ω, ω^{'}) = [P (\frac{1}{ω - ω^{'}}) + γ_{k} (ω) δ (ω - ω^{'})] V_{k}^{*} (ω^{'}) \frac{{\bar{ω}}_{k}}{ω + {\bar{ω}}_{k}} x_{k} (ω), \\ w_{k} (ω, ω^{'}) = [\frac{1}{ω + ω^{'}}] V_{k} (ω^{'}) \frac{{\bar{ω}}_{k}}{ω + {\bar{ω}}_{k}} x_{k} (ω) . \end{gathered}

(B8)

The function γ_k(ω) can be found, after some algebra, replacing both the equations in Eq. (B8) with Eq. (B3),

γ_{k} (ω) = \frac{2 (ω^{2} - {\bar{ω}}_{k}^{2})}{{\bar{ω}}_{k} | V_{k} (ω) |^{2}} + \frac{1}{| V_{k} (ω) |^{2}} P \int_{- \infty}^{\infty} d ω^{'} \frac{V_{k} (ω^{'})}{ω^{'} - ω},

(B9)

where we assume that the analytic extension in the negative frequency range $V_{k} (ω)$ of |V_k(ω)|² is an odd function. In order to calculate x_k(ω), we impose the commutation relation

[A_{k} (ω), A_{k^{'}}^{†} (ω^{'})] = δ_{k, k^{'}} δ (ω - ω^{'}) .

(B10)

By using the expression for A_k(ω) in Eq. (B1) and for the coefficients in Eqs. (B4) and (B8), in terms of x_k(ω), Eq. (B10) leads to the definition of x_k(ω) up to a phase factor,

x_{k} (ω) = \frac{ω + {\bar{ω}}_{k}}{\bar{ω} V_{k}^{*} (ω)} \frac{1}{γ_{k} (ω) - i π} .

(B11)

Exploiting the expression for γ_k(ω) in Eq. (B9), Eq. (B11) can be written as

x_{k} (ω) = \frac{ω + {\bar{ω}}_{k}}{2} \frac{V_{k} (ω)}{ω^{2} - {\bar{ω}}_{k}^{2} χ_{k} (ω)}

(B12)

with

χ_{k} (ω) = 1 - \frac{1}{2 {\bar{ω}}_{k}} \int_{- \infty}^{\infty} d ω^{'} \frac{V_{k} (ω^{'})}{ω^{'} - ω + i 0^{+}} .

(B13)

The final expressions for the coefficients are thus obtained as

\begin{gathered} x_{k} (ω) = \frac{ω + {\bar{ω}}_{k}}{2} \frac{V_{k} (ω)}{ω^{2} - {\bar{ω}}_{k}^{2} χ_{k} (ω)}, \\ z_{k} (ω) = \frac{ω - {\bar{ω}}_{k}}{2} \frac{V_{k} (ω)}{ω^{2} - {\bar{ω}}_{k}^{2} χ_{k} (ω)}, \\ y_{k} (ω, ω^{'}) = δ (ω - ω^{'}) + \frac{{\bar{ω}}_{k}}{2} \frac{V_{k} (ω^{'})}{ω - ω^{'} - i 0^{+}} \frac{V_{k} (ω)}{ω^{2} - {\bar{ω}}_{k}^{2} χ_{k} (ω)}, \\ w_{k} (ω, ω^{'}) = \frac{{\bar{ω}}_{k}}{2} \frac{V_{k} (ω^{'})}{ω + ω^{'}} \frac{V_{k} (ω)}{ω^{2} - {\bar{ω}}_{k}^{2} χ_{k} (ω)} . \end{gathered}

(B14)

We have thus demonstrated that the operators A_k(ω) are eigensolutions of H_PB. The proof these operators form a complete set of solutions in the case of an unbounded continuum can be found in the original HB paper³ in which the linear transformation in Eq. (B1) is inverted, expressing the discrete a_k operator as a linear superposition of A_k(ω).

APPENDIX C: DIAGONALIZATION IN THE COULOMB REPRESENTATION

In this appendix, we show how our approach gets modified if one wants to work in the Coulomb representation. Given that only relatively few quantities are affected by the change, we just provide the expressions for the affected ones. The Hamiltonian in the Coulomb representation can be written as

\begin{align} H & = \sum_{k} (ω_{k} a_{k}^{†} a + ω_{x} b_{k}^{†} b_{k}) + i \sum_{k} g \sqrt{\frac{ω_{x}}{ω_{k}}} [a_{k}^{†} + a_{k}] [b_{k}^{†} - b_{k}] \\ + \sum_{k} \frac{| g |^{2}}{ω_{k}} {[a_{k}^{†} + a_{k}]}^{2}, \end{align}

(C1)

in which we can see the appearance of a diamagnetic A² term. After reabsorbing such a diamagnetic term by performing a Bogoliubov transformation, the Hamiltonian takes the form

H = \sum_{k} ({\tilde{ω}}_{k} {\tilde{a}}_{k}^{†} \tilde{a} + ω_{x} b_{k}^{†} b_{k}) + i \sum_{k} \sqrt{\frac{ω_{x}}{{\bar{ω}}_{k}}} [{\tilde{a}}_{k}^{†} + {\tilde{a}}_{k}] [g b_{k}^{†} - g^{*} b_{k}]

(C2)

with rotated photon operators

{\tilde{a}}_{k} = \frac{{\tilde{ω}}_{k} + ω_{k}}{2 \sqrt{{\tilde{ω}}_{k} ω_{k}}} a_{k} + \frac{{\tilde{ω}}_{k} - ω_{k}}{2 \sqrt{{\tilde{ω}}_{k} ω_{k}}} a_{k}^{†}

(C3)

and renormalized cavity frequency

{\tilde{ω}}_{k}^{2} = ω_{k}^{2} + 4 | g_{k} |^{2} .

(C4)

The dispersion relation obtained diagonalizing the Hamiltonian in Eq. (C2) is

ω_{\pm, k}^{2} - {\tilde{ω}}_{k}^{2} = \frac{4 | g |^{2} ω_{x}^{2}}{ω_{\pm, k}^{2} - ω_{x}^{2}}

(C5)

with solutions

ω_{\pm, k} = \frac{1}{\sqrt{2}} \sqrt{{\tilde{ω}}_{k}^{2} + ω_{x}^{2} \pm \sqrt{{({\tilde{ω}}_{k}^{2} - ω_{x}^{2})}^{2} + 16 | g |^{2} ω_{x}^{2}}} .

(C6)

In the Coulomb representation, the expressions of the field operators in Eq. (29) are modified as

\begin{gathered} ({\tilde{a}}_{k} + {\tilde{a}}_{k}^{†}) = \sqrt{{\bar{ω}}_{k}} \int_{0}^{\infty} d ω [ζ_{k} (ω) A_{k}^{†} (ω) + ζ_{k}^{*} (ω) A_{k} (ω)], \\ i (b_{k}^{†} - b_{k}) = \frac{1}{\sqrt{{\bar{ω}}_{x}}} \int_{0}^{\infty} d ω^{'} [η (ω) B_{k}^{†} (ω) + η^{*} (ω) B_{k} (ω)], \end{gathered}

(C7)

where

\begin{gathered} ζ_{k} (ω) = \frac{1}{\sqrt{{\bar{ω}}_{k}}} [x_{k} (ω) - z_{k} (ω)] = \frac{V_{k} (ω) \sqrt{{\bar{ω}}_{k}}}{ω^{2} - {\bar{ω}}_{k}^{2} χ_{k} (ω)}, \\ η (ω) = i \sqrt{{\bar{ω}}_{x}} [\bar{x} (ω) + \bar{z} (ω)] = i \frac{Q (ω) ω \sqrt{{\bar{ω}}_{x}}}{ω^{2} - {\bar{ω}}_{x}^{2} t (ω)} . \end{gathered}

(C8)

By substituting the bare operators with the dressed ones, we arrive at the Hamiltonian as in Eq. (31), which can be diagonalized by the same procedure described in the main text. The dressed photonic and matter field operator can be finally written as superpositions of polaritonic broadened modes as

\begin{gathered} ({\tilde{a}}_{k} + {\tilde{a}}_{k}^{†}) = \sqrt{{\bar{ω}}_{k}} \int_{0}^{\infty} d ω \sum_{j = \pm} [K_{k, j}^{*} (ω) P_{j} (ω) + K_{k, j} (ω) P_{j}^{†} (ω)], \\ i (b_{k} - b_{k}^{†}) = \frac{1}{\sqrt{{\bar{ω}}_{x}}} \int_{0}^{\infty} d ω \sum_{j = \pm} [J_{k, j}^{*} (ω) P_{j} (ω) + J_{k, j} (ω) P_{j}^{†} (ω)], \end{gathered}

(C9)

where

\begin{gathered} K_{k, j} (ω) = \frac{1}{\sqrt{{\bar{ω}}_{k}}} [X_{k, j} (ω) - Z_{k, j} (ω)], \\ J_{j} (ω) = - i \sqrt{{\bar{ω}}_{x}} [Y_{j} (ω) + W_{j} (ω)] . \end{gathered}

(C10)

Note that in the Coulomb representation, the functions related to the photonic component and those related to the matter part have exchanged units from those in the PZW representation. This is due to the inverted dependence of the light and matter fields upon their frequency.

APPENDIX D: DERIVATION OF ANALYTICAL RESULTS FOR A LORENTZIAN BROADENING

In this appendix, we will derive the analytical expression of the functions ζ_k(ω) and η(ω) in the case of a Lorentzian broadening. This is not completely trivial, as testified by the existence of a published paper claiming it is impossible.⁴¹ The key issue is that in order to recover a frequency-independent broadening, the shift in the mode due to the coupling with the reservoir has to diverge. As such, the result can only be found by a renormalization procedure allowing to cancel such a divergence.

Assuming that the coupling potentials to the photonic and matter reservoirs take the form in Eq. (50), we can calculate the dressed resonance frequencies

\begin{gathered} {\bar{ω}}_{k}^{2} = ω_{k}^{2} + \int_{0}^{\infty} d ω \frac{| V_{k} (ω) |^{2} {\bar{ω}}_{k}}{ω} = ω_{k}^{2} \frac{q_{P} + ω_{P}}{q_{P}}, \\ {\bar{ω}}_{x}^{2} = {\tilde{ω}}_{x}^{2} + \int_{0}^{\infty} d ω \frac{| Q (ω) |^{2} {\bar{ω}}_{x}}{ω} = {\tilde{ω}}_{x}^{2} \frac{q_{M} + ω_{M}}{q_{M}}, \end{gathered}

(D1)

and the real and imaginary parts of functions χ_k(ω) and t(ω) as

\begin{gathered} \begin{aligned} Re [χ_{k} (ω)] & = \lim_{ω_{P} \to \infty} [1 - \frac{1}{2} \frac{1}{q_{P} + ω_{P}}) \\ (\times (2 ω_{P} + 2 ω \log (1 - \frac{2 ω}{ω + ω_{P}}))], \end{aligned} \\ Im [χ_{k} (ω)] = \lim_{ω_{P} \to \infty} \frac{π ω}{2 (q_{P} + ω_{P})}, \\ \begin{aligned} Re [t (ω)] & = \lim_{ω_{M} \to \infty} [1 - \frac{1}{2} \frac{1}{q_{M} + ω_{M}}) \\ \times ((2 ω_{M} + 2 ω \log (1 - \frac{2 ω}{ω + ω_{M}}))], \end{aligned} \\ Im [t (ω)] = \lim_{ω_{M} \to \infty} \frac{π ω}{2 (q_{M} + ω_{M})} . \end{gathered}

(D2)

In the limit of infinite cut-off frequency ω_P → ∞ and ω_M → ∞, the resonant frequencies diverge, but the intensity of the coupling vanishes, and we arrive at the finite results,

\begin{gathered} \lim_{ω_{P} \to \infty} {\bar{ω}}_{k}^{2} Re [χ_{k} (ω)] = {\bar{ω}}_{k}^{2} (\frac{q_{P}}{q_{P} + ω_{P}}) = ω_{k}^{2}, \\ \lim_{ω_{P} \to \infty} {\bar{ω}}_{k}^{2} Im [χ_{k} (ω)] = ω_{k}^{2} \frac{π ω}{2 q_{P}} . \\ \lim_{ω_{M} \to \infty} {\bar{ω}}_{x}^{2} Re [t (ω)] = {\tilde{ω}}_{0}^{2} (\frac{q_{M}}{q_{M} + ω_{M}}) = {\tilde{ω}}_{x}^{2}, \\ \lim_{ω_{M} \to \infty} {\bar{ω}}_{x}^{2} Im [t (ω)] = {\tilde{ω}}_{x}^{2} \frac{π ω}{2 q_{M}} . \end{gathered}

(D3)

By inserting Eq. (D3) into Eq. (30), we finally obtain the Lorentzian form for the functions ζ_k(ω) and η(ω) as

ζ_{k} (ω) = i \sqrt{\frac{2 γ_{P} ω^{3}}{π}} \frac{1}{ω^{2} - ω_{k}^{2} - i γ_{P} ω},

(D4)

η (ω) = g \sqrt{\frac{2 γ_{M} ω}{π}} \frac{1}{ω^{2} - {\tilde{ω}}_{x}^{2} - i γ_{M} ω} .

(D5)

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