Cavity Quantum Electrodynamics with Frequency-Dependent Reflectors
Ondřej Černotík Aurélien Dantan Claudiu Genes












Present address: Department of Optics, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
ondrej.cernotik@upol.cz
Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany



Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark



Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany



    We present a general framework for cavity quantum electrodynamics with strongly frequency-dependent mirrors. The method is applicable to a variety of reflectors exhibiting sharp internal resonances as can be realized, for example, with photonic-crystal mirrors or with two-dimensional atomic arrays around subradiant points. Our approach is based on a modification of the standard input–output formalism to explicitly include the dynamics of the mirror's internal resonance. We show how to directly extract the interaction tuning parameters from the comparison with classical transfer matrix theory and how to treat the non-Markovian dynamics of the cavity field mode introduced by the mirror's internal resonance. As an application within optomechanics, we illustrate how a non-Markovian Fano cavity possessing a flexible photonic-crystal mirror can provide both sideband resolution as well as strong heating suppression in optomechanical cooling. This approach, amenable to a wide range of systems, opens up possibilities for using hybrid frequency-dependent reflectors in cavity quantum electrodynamics for engineering novel forms of light–matter interactions.





Cavity Quantum Electrodynamics with Frequency-Dependent Reflectors
    Claudiu Genes
    December 30, 2023
==================================================================













A standard platform for cavity quantum electrodynamics (CQED) <cit.> is the linear Fabry–Pérot resonator;
one generally assumes two macroscopic, highly reflecting mirrors that define spatially-localized frequency-resolved resonances inside the cavity.
A full quantum description of the cavity mode dynamics can be derived in the form of a Langevin equation ȧ(t) = -iω_a a(t)-κ a(t)+√(2κ) a_(t)
where a is the annihilation operator of the field mode with frequency ω_a and decay rate κ and a_(t) describes delta-correlated input noise encompassing the effect of the coupling to the continuum of outside modes <cit.>. The solution, combined with the input–output relation a_(t)=a_(t) -√(2κ) a(t), describes the quantum properties of the continuous outgoing light field a_(t).
In such a case, the quantum dynamics of the cavity field are Markovian, the coupling to the continuum of outside modes giving rise to an exponential time decay of the intracavity field.
A critical step in this derivation lies in assuming that the reflectivity of the mirrors is essentially flat around the resonance frequency of interest.

Many scenarios, however, strongly depart from this situation as end-mirrors can be made of reflective materials exhibiting enhanced linear or nonlinear response around frequencies corresponding to sharp internal modes (Fig. <ref>).
In two-dimensional systems, these effects can be achieved by patterning a subwavelength grating or a photonic-crystal structure onto a dielectric membrane <cit.>;
other systems can be formed by semiconducting monolayers <cit.> or two-dimensional arrays of atoms trapped in optical lattices <cit.>.
Using such metamaterials with a strongly frequency-dependent response as end-mirrors in Fabry–Pérot resonators has been shown to result in asymmetric transmission profiles potentially much narrower than those obtained with frequency-independent mirrors of comparable reflectivity <cit.> (Fig. <ref>(c)).


This manuscript addresses provides a generalized approach to CQED with mirrors possessing sharp internal resonances.
Our strategy is based on an extension of the standard derivation of cavity input–output relations that includes, as an additional degree of freedom, the quantum dynamics of the mirror's internal mode and its coupling to both the cavity field and the continuum outside (Fig. <ref>).
The result is a compact Langevin equation for the cavity field,

    ȧ(t) = -iω_a a(t)-(κ_∗ a)(t)+(κ_∗ a_)(t),

with a generalized decay rate and input noise encompassing [via the convolution (f∗ g)(t)] non-Markovian effects stemming from the delayed response of the field to the input noise. The explicit dependence of κ_(t) and κ_(t) on the system parameters is obtained by identifying the predicted transmission of the quantum model to classical results obtained via a transfer matrix approach (simulation of Maxwell's equations in one dimension). We illustrate this procedure on the particular example of a cavity with a photonic-crystal membrane (Fano mirror) by using a generic frequency-dependent polarizability derived for this system by Fan et al. <cit.>. We then use this quantum coupled modes approach to illustrate a non-Markovian regime for cavity optomechanics (OM) where the mirror-induced reduction in the cavity linewidth combined with the suppression of the heating sideband by the asymmetric cavity profile can provide enhanced cooling dynamics.


  
Non-Markovian Langevin equations.
A standard quantum optics approach <cit.> to deriving Langevin equation of motion for a mode a of an optical cavity with leaky mirrors starts from the complete Hamiltonian H = H_S + H_B + of the system, bath and their interaction. The free evolution is given by H_S = ω_a a^† a (for system), H_B = ∫ dω ω[b^†(ω)b(ω)+c^†(ω)c(ω)] (accounting for the external free fields b(ω), c(ω)) while the mutual coupling is = ∫ dω i[V_ω b(ω)+W_ω c(ω)]a^† +.  The model assumes an exchange interaction with frequency-dependent rates V_ω and W_ω describing tunnelling of photons between the cavity and the free fields through the end-mirrors. Formal integration of the evolution of the outside modes leads to an effective dissipative dynamics for the cavity mode a. A critical step is the simplifying assumption that V_ω and W_ω are weakly dependent on frequency such that only their value at the cavity resonance is relevant.
 For the situation depicted in Fig. <ref>(a) the right-hand mirror is frequency-independent, ensuring that W_ω is practically constant around the resonance. The same is however not true for the sharply peaked frequency response of the left-hand mirror. Therefore, the coupling between a and the mirror-filtered outside modes V_ω is necessarily strongly frequency-dependent. The elimination of the outside modes strictly following the standard formalism is therefore not straightforward. However, the approach of Ref. <cit.> suggests that the task can be greatly simplified by explicitly including the time dynamics of the left-hand mirror's internal mode.
This modification is done by adding a quantized mode d directly coupled to the cavity and to the outside continuum.
We adjust the free Hamiltonian to include the free evolution ω_d d^† d of mode d at frequency ω_d and its coupling to the cavity mode λ(a^† de^iϕ+d^† ae^-iϕ) with complex strength λ e^iϕ. The interaction Hamiltonian is also modified by a term U_ω b(ω)d^† + , in order to provide a mechanism for direct exchange of photons between the mirror mode and the left-side modes b(ω) at rate U_ω. As all bosonic modes are now directly coupled to their baths, it is now allowed to consider the tunnelling rates U,V,W∈ℝ to be frequency independent (hence, we drop the subscript ω). Tunneling will then give rise to loss rates κ_L = π V^2, κ_R = π W^2, γ = π U^2.
The dynamics are then described by the Langevin equations (see Supplemental Material-SM <cit.>)


    ȧ   = -(iω_a+κ)a - 𝒢_+d + √(2κ_L)b_ + √(2κ_R)c_, 
    ḋ   = -(iω_d+γ)d - 𝒢_-a + √(2γ)b_,


where we introduced 𝒢_± = iλ e^± iϕ+√(κ_Lγ), the total decay rate κ = κ_L+κ_R and the input fields on the left and right denoted by b_ and c_.
The input fields have the usual non-vanishing correlation functions b_(t)b_^†(t') = δ(t-t') (and similarly for c_(t)), while all other correlations vanish.
The associated output fields follow the input–output relations b_ = b_-√(2κ_L)a-√(2γ)d, and c_ = c_ -√(2κ_R)a. As we are interested in the dynamics of the cavity mode alone, we can formally integrate the evolution of the mirror mode
and replace it back in Eq. (<ref>).
We thus recover the non-Markovian cavity dynamics of Eq. (<ref>) where we can identify the memory kernels


    κ_(t)    = 2κ_Lδ(t)-𝒢_+𝒢_-e^-(iω_d+γ)t,
    κ_(t)    = 2√(2κ_L)δ(t)-𝒢_+√(2γ)e^-(iω_d+γ)t.


Notice that Eq. (<ref>) is written for a single input noise. The double-sided cavity considered here exhibits a Markovian response to c_ and a non-Markovian response to b_ characterized by the effective time-dependent decay rate κ_(t) and associated response with  κ_(t) to the left-side input noise field b_.
The outgoing field b_ will also show a memory effect as it includes a time-delayed response via the mirror mode of both a(t) and b_(t) according to


    b_   = (κ_∗ b_)(t) - (κ_'∗ a)(t),
    κ_(t)    = 2δ(t)-2γ e^-(iω_d+γ)t,
    κ_'(t)    = 2√(2κ_L)δ(t)-𝒢_-√(2γ)e^-(iω_d+γ)t.


The expressions above are completely general as they do not involve any assumptions on the particular nature of the mirror's internal mode. The parameters λ,κ_L,κ_R,ω_a are to be determined from comparisons of the predictions of the coupled mode quantum model for cavity transmission, reflection with classical transfer matrix methods. In general, κ_L is not the dissipation rate of the cavity quasi-mode but rather a very large rate of coupling to the vacuum modes; also, the cavity quasi-mode frequency will be defined by the resonance of the mirror. Furthermore, against intuition, the limit of frequency-insensitive mirrors, reached for infinite γ leads to infinite couplings 𝒢_±. However, as we will illustrate with photonic crystal mirrors, terms such as γ/2 e^-γ |t| tend to δ(t) for infinite γ, thus reproducing the expected instantaneous κ_(t)∝δ(t) response characteristic of Markovian dynamics.


  
Transfer matrix comparison.
We now describe a roadmap to extract the parameters of the coupled-mode model by matching its predictions to a classical transfer matrix calculation. The transmission coefficient t(ω)=c_(ω)/b_(ω)=√(κ_R)a(ω)/b_(ω) can be analytically obtained by solving the equations of motion (<ref>) in the frequency domain such that


    t(ω) = 2i√(κ_R)[e^iϕλ√(γ)-(ω_d-ω)√(κ_L)]/λ^2+κ_Rγ-(ω_d-ω)(ω_a-ω)+i[γ(ω_a-ω)+κ(ω_d-ω)-2λ√(κ_Lγ)cosϕ].


The transfer matrix approach, on the other hand, consists in solving the classical one-dimensional wave propagation equation in a one-dimensional setup with two mirrors parametrized by polarizabilities ζ_R and ζ_L(ω).
In linear response theory, one can find the transmission function of the setup for any incoming plane wave at a given frequency ω.
As detailed in SM <cit.>, we get the classical transmission coefficient

    t̃(ω)
        = 1/(1-iζ_R)[1-iζ_L(ω)]e^-iθ+ζ_Rζ_L(ω)e^iθ;

here, θ = 1/2[(ω_d-ω)/Γ + arctan(1/ζ_R)+arctan(1/ζ_0)] with Γ = c/2L denoting the free spectral range and ζ_0 = ζ_L(ω = ω_d).
Comparison of the two expressions, t(ω) and t̃(ω), gives the free parameters of the coupled mode description. We will exemplify this procedure below in the particular case of a mirror formed by a photonic-crystal membrane.





  
Photonic crystal mirrors.
Fano resonances in thin dielectric membranes patterned with subwavelength photonic-crystal structures may generically be described by the polarizability <cit.>

    ζ_L(ω) = ζ_0γ-2s(ω_d-ω)/ζ_0/γ+2sζ_0(ω_d-ω),

with s specifying the orientation of the Fano resonance [see Fig. <ref>(a)].
We assume ζ_0≫ 1, leading to a small off-resonant value ζ_L→ 1/ζ_0 at ω→±∞.
The left mirror is, therefore, a poor reflector of its own and only the presence of its internal mode gives rise to a large effective reflectivity on resonance.
To extract the parameters of the coupled mode quantum theory from the classical transfer matrix results we follow three steps: i) identifying the zero of the cavity transmission , ii) taking the limit γ≫Γ (Markovian cavity) and iii) taking the limit of fully non-Markovian behavior with γ≪Γ. The three conditions lead to <cit.>

    λ = -s√(κ_0γ),    ω_a = ω_d -2s√(κ_0κ_L).

The decay rates can be identified as  κ_0 = Γ/2ζ_0^2, κ_L = 2Γ, and κ_R = Γ/2ζ_R^2 and ϕ = 0. Notice that κ_0 and κ_R are the expected loss rates of photons to the continuum for frequency-independent mirrors with fixed ζ_0,ζ_R. The quantity κ_L instead describes the high loss of the cavity to the outside modes: far from the mirror resonance the effective finesse is proportional to ζ_Rζ_L(ω→±∞)=ζ_R/ζ_0, which is of order unity and describes a quasi-mode with linewidth of the order of the free spectral range.

In the limit γ≫Γ Markovian dynamics is expected. With 𝒢_±=𝒢=-is√(κ_0γ)+√(κ_Lγ) one can now compute the terms appearing in Eq. (<ref>) and take the limit γ→∞ <cit.>. We obtain κ_(t) = 2κ_0δ(t)-4is√(κ_0κ_L)δ(t), which, after time integration, leads to the expected result that the mode a is shifted by the quantity 2s√(κ_0κ_L) to fit the mirror frequency ω_d and effectively decays at rate κ_0 through the left mirror (in addition to the decay at rate κ_R through the frequency independent right-mirror). The expected Lorentzian spectrum with width κ_0+κ_R is then observed [Fig. <ref>(b)].

In the limit γ≪Γ the Fano mirror leads a narrower cavity transmission profile with effective linewidth κ_ = γ/ζ_0^2 much smaller than the linewidth Γ/ζ_0^2 of a standard cavity with fixed ζ_0. Moreover, the Fano interference manifests itself in a non-Markovian asymmetry with a zero in the transmission, tunable in position by the design of the photonic crystal (s = ± 1). Both effects are illustrated in Fig. <ref>(b). Perfect agreement between exact transfer matrix simulations and the coupled mode theory is obtained expect in an intermediate regime γ∼Γ where the mirror's reflectivity spectrum becomes broad enough to interact with multiple cavity modes <cit.>. However, since the transmission profile becomes much broader we do not focus on this regime.



  
Optomechanical sideband cooling.
Let us analyze the performance of a cavity OM setup with frequency-dependent mirrors in the non-Markovian regime with γ≪Γ. Conventional OM employs a radiation pressure Hamiltonian, H_RP = g_0 a^† aq (where q is the dimensionless position quadrature of the vibrating mirror), which, in the standard weak phonon-photon coupling, can be transformed into a linearized interaction H_lin = g(a+a^†)q <cit.>.
The coupling rate g = g_0α is enhanced by the field amplitude in steady state α≫ 1.
A decomposition into creation and annihilation operators, q = (b+b^†)/√(2), shows two contributions: i) a beam-splitter interaction g(a^† b + b^† a) responsible for state transfer or cooling and ii) a two-mode squeezing interaction g(ab+a^† b^†) resulting in entanglement or heating.
When the frequency of the driving laser ω_L fulfills ω_L=ω_a- ω_m [see Fig. <ref>(a,c)], the beam-splitter interaction is resonant and the cavity field serves as an additional zero-temperature bath for the mechanical resonator; phonons are converted into photons which leave the cavity through its end-mirrors.
The heating contribution sets a lower limit to achievable final occupancies which scales with the sideband ratio κ/ω_m, such that ground state cooling is more efficient in the resolved sideband regime, κ<ω_m <cit.>.

An inspection of the cavity transmission profile in Fig. <ref>(b) reveals two advantages of using frequency-dependent reflectors: i) the reduced linewidth enables a better sideband resolution and ii) the interference between the mirror and cavity modes leads to vanishing density of modes in a small frequency window that can result in the suppression of Stokes scattering. To assess these expectations, we derive perturbative expressions for the cooling rate and final occupancy starting from the Langevin equations for the mechanical resonator,

    q̇ = ω_m p,    ṗ = -ω_m q - γ_m p + f_th - F,

which describe its free evolution at frequency ω_m and damping at a rate γ_m;
f_th, with the correlation function f_th(t)f_th(t') = γ_m(2+1)δ(t-t'), is the associated thermal noise operator.

Cooling from an initial high temperature (characterized by the average thermal occupation ≫ 1) is achieved by the Langevin radiation pressure force F = g(a+a^†); its effect can be quantified using the cooling rate Γ_cool = 1/2[S_F(ω_m)-S_F(-ω_m)]
with S_F(ω) denoting the spectral density of F at frequency ω. We find <cit.>


    S_F(ω) = 2g^2κ_R[γ^2+(ω-Δ_d)^2]+2g^2[γ√(κ_0)-s√(κ_L)(ω-Δ_d)]^2/[(κ+γ)(ω-Δ_d)]^2 + [γ(κ_0+κ_R)-(ω-Δ_d)(ω-Δ_d+2s√(κ_0κ_L))]^2.


Heating is minimized for Δ_d = ω_d-ω_L = -ω_m-sγ√(κ_0/κ_L) = -ω_m -sγ/2ζ_0; its rate is S_F(-ω_m)=2g^2 κ_Rζ_0^2/Γ^2 and vanishes for a one-sided cavity (with κ_R = 0).
Cooling is maximized for the detuning Δ_d = ω_m at the rate S_F(ω_m) = 2g^2/(κ_0+κ_R).
Ideally, we will both minimize the Stokes and maximize the anti-Stokes scattering;
this can be achieved for s = -1 and optimal mirror linewidth γ_ = 4ζ_0ω_m, as illustrated in Fig. <ref>(b).
Note that one can, similarly, suppress the anti-Stokes scattering for Δ_d = -ω_m, with the interaction phase s = +1. Such a setting might offer an advantage when strong amplification of the mechanical motion is necessary, for example for reaching mechanical limit cycles <cit.>. The comparison with exact numerical simulations <cit.> is presented in Fig. <ref>(c). First, we plot the final occupancy for standard cooling in a Markovian cavity with the left mirror characterized by a fixed ζ_0 (dotted line). Then, we consider a non-Markovian cavity with a Fano mirror having γ≪Γ and observe that the cooling performance is greatly enhanced by a combination of linewidth narrowing and non-Markovian cancellation of heating rates (full lines). The Fano effect of Stokes suppression is evident when comparing with a Markovian cavity with the same narrow linewidth κ_ = γ/2ζ_0^2.
The non-Markovian cavity exhibits a smaller cooling rate ∝1/(κ_0+κ_R) than the equivalent Markovian one ∝1/κ_. The advantage lies in the much lower final occupancies achievable owing to the suppression of heating.


  
Discussion.
Besides sideband cooling the reduced Stokes scattering obtained with frequency-dependent reflectors can also improve the fidelity of state transfer <cit.> and frequency conversion in optomechanical systems <cit.>.
Suppression of anti-Stokes scattering can be beneficial for more efficient amplification of electromagnetic fields or mechanical motion <cit.> and for reaching mechanical limit cycles <cit.>.
The reduced cavity linewidth can enable all these applications also in micromechanical cavities <cit.>, which are typically precluded from reaching the resolved sideband regime owing to their large linewidths. Photonic crystal membranes received attention lately as a possible platform for quantum OM <cit.>;
we show how the effects that may arise owing to their frequency-dependent reflectivity can be rigorously described quantum mechanically. Our results can furthermore be applied to other platforms as well, such as arrays of trapped atoms <cit.> or semiconductor membranes <cit.>.

In the future, it would be interesting to analyze how cavity QED with quantum emitters is modified by using frequency-dependent end-mirrors.
Here, the modified width and shape of the cavity mode is expected to lead to nontrivial dynamics of strongly coupled polaritons or modified photon statistics. Coupling ensembles of quantum emitters to non-Markovian cavities could as well lead to modified super- or subradiance <cit.> and lasing <cit.>. Frequency-dependent reflection has also been employed in photonic crystals with quantum dots <cit.>
and it can be used in one-dimensional systems with emitters in optical waveguides <cit.> and in superconducting <cit.> or plasmonic systems <cit.>.








    We gratefully acknowledge financial support from the Max Planck Society and the Velux Foundations. We thank Christian Sommer for relevant comments on the manuscript.


























    Supplemental Material: Cavity Quantum Electrodynamics with Frequency-Dependent Reflectors










§ COUPLED-MODE THEORY FOR CAVITIES WITH FREQUENCY-DEPENDENT REFLECTORS


Here, we present a derivation of the Langevin equations and input–output relations for the coupled mode description of a cavity with a resonant end mirror.
We start from the full Hamiltonian describing the dynamics of the cavity field, the mirror mode, and the two continua to the left and right of the cavity H = H_S+H_B+ as presented in the main text:


    H_S    = ω_a a^† a+ω_d d^† d + λ(a^† de^iϕ+d^† ae^-iϕ),
    
        H_B    = ∫ dω ω[b^†(ω)b(ω)+c^†(ω)c(ω)],
       = ∫ dω i[V_ω b(ω)+W_ω c(ω)]a^† + U_ω b(ω)d^† +


Next, we formulate the equations of motion for the four fields,


    ȧ   = -iω_a a-iλ e^ iϕd + ∫ dω [V_ω b(ω)+W_ω c(ω)],
    ḋ   = -iω_d d-iλ e^-iϕa + ∫ dω U_ω b(ω),
    ḃ(ω)    = -iω b(ω) - V_ω a - U_ω d,
    ċ(ω)    = -iω c(ω) - W_ω a,


and formally integrate the equations for the two continua,


    b(ω)    = e^-iω(t-t_0)b_0(ω) -∫_t_0^t dτ e^-iω(t-τ)(V_ω a+U_ω d),
    
        c(ω)    = e^-iω(t-t_0)c_0(ω) -∫_t_0^t dτ e^-iω(t-τ)W_ω a,


where we assume the initial conditions b_0(ω), c_0(ω) at time t_0→ -∞.

Now, we plug the results (<ref>) back to the equations of motion for the cavity field and mirror mode.
We assume that the response of the mirrors is sufficiently flat for the frequencies of interest so that we can assume the tunneling rates U,V,W to be frequency independent.
We can then evaluate the integrals over frequency in Eqs. (<ref>), (<ref>) (each of which yields a delta function) and obtain


    ȧ   = -(iω_a +κ_L+κ_R)a -(iλ e^iϕ+√(κ_Lγ))d + √(2κ_L)b_ + √(2κ_R)c_
    ḋ   = -(iω_d+γ)d -(iλ e^-iϕ+√(κ_Lγ))a + √(2γ)b_;


here, we introduced the decay rates κ_L = π V^2, κ_R =π W^2, γ =π U^2 and the input field

    b_ = 1/√(2π)∫ dω  e^-iω(t-t_0)b_0(ω);

an analogous definition holds also for c_.

Alternatively, the equations of motion for the two continua can be solved using the final conditions b_1(ω), c_1(ω) at time t_1→∞,


    b(t,ω)    = e^-iω(t-t_1)b_1(ω) +∫_t^t_1 dτ e^-iω(t-τ)(V_ω a+U_ω d),
    
        c(t,ω)    = e^-iω(t-t_1)c_1(ω) +∫_t^t_1 dτ e^-iω(t-τ)W_ω a.


Introducing the output field

    b_ = 1/√(2π)∫ dω  e^-iω(t-t_1)b_1(ω)

and expressing

    1/√(2π)∫ dω b(ω)    = b_ - √(κ_L/2)a - √(2/γ)d 
       = b_ +√(κ_L/2)a + √(2/γ)d,

we obtain the input–output relations


    b_   = b_ - √(2κ_L)a - √(2γ)d,
    
        c_   = c_ - √(2κ_R)a.


The input–output relation (<ref>) can be derived in full analogy to Eq. (<ref>).
Note that, in the equations above, the description of a standard two-sided cavity can be recovered by putting λ = U = γ = 0.

Non-Markovian dynamics of the cavity field can now be obtained by solving the equation for the mirror mode formally,

    d(t)=∫_-∞^tdτ  e^-(iω_d+γ)(t-τ)[√(2γ)b_(τ)- 𝒢_-a(τ)],

and plugging this solution into the equation of motion for the cavity field, Eq. (<ref>), and the input–output relation (<ref>).
We obtain


    ȧ   = -iω_a a -κ_R a+√(2κ_R)c_ -∫_-∞^t dτ [κ_(t-τ)a(τ)+ κ_(t-τ)b_(τ)], 
    
        b_   = ∫_-∞^t dτ [κ_(t-τ)b_(τ) - κ_'(t-τ)a(τ)];


the coefficients κ_, κ_, κ_', κ_ are given by


    κ_(t)    = 2κ_Lδ(t)-𝒢_+𝒢_-e^-(iω_d+γ)t
    κ_(t)    = 2√(2κ_L)δ(t)-𝒢_+√(2γ)e^-(iω_d+γ)t,
    κ_'(t)    = 2√(2κ_L)δ(t)-𝒢_-√(2γ)e^-(iω_d+γ)t,
    κ_(t)    = 2δ(t)-2γ e^-(iω_d+γ)t.



For photonic crystal membrane cavities, we derive below


    𝒢_+    = 𝒢_- = 𝒢 = √(κ_Lγ) -is √(κ_0γ),
    ω_a    = ω_d-2s√(κ_0κ_L).


We can use these expressions to obtain the effective time-dependent decay rates


    κ_(t)    = 2κ_Lδ(t)-𝒢^2K(t),
    κ_(t) = κ_'(t)    = 2√(2κ_L)δ(t)-𝒢√(2γ)K(t),
    κ_(t)    = 2δ(t)-2γ K(t),


where we introduced the kernel function K(t) = exp[-(iω_d+γ)t].
In the limit γ→∞, we have γ e^-γ t→ 2δ(t) so the rates defined in Eqs. (<ref>) become proportional to the delta function.
We thus recover the expected Markovian dynamics


    ȧ   = -(iω_d+κ_R+κ_0) a + i√(2κ_0)b_ + √(2κ_R) c_,
    
        b_   = -b_ +i√(2κ_0)a;


the resonance frequency of the cavity is shifted from ω_a to ω_d in agreement with the transfer matrix calculation (see the next section).








§ COMPARISON TO TRANSFER MATRIX DESCRIPTION


In this section, we show how the parameters of the coupled-mode model can be related to experimentally accessible quantities.
We start by evaluating the transmission function from the coupled-mode theory.
First, we solve the equations of motion (<ref>) in the frequency domain; this yields the result


    a(Δ)    = [(γ+iΔ)√(2κ_L)-𝒢_+√(2γ)]b_+(γ+iΔ)√(2κ_R)c_/(κ+iΔ+iδ)(γ+iΔ)-𝒢_+𝒢_-,
    
        d(Δ)    = [(κ+iΔ+iδ)√(2γ)-𝒢_-√(2κ_L)]b_-𝒢_-√(2κ_R)c_/(κ+iΔ+iδ)(γ+iΔ)-𝒢_+𝒢_-


with the detunings Δ = ω_d-ω, δ = ω_a-ω_d.
Next, we plug Eqs. (<ref>) into the input–output relations (<ref>).
We thus obtain an expression that describes the transmission from the input field b_ to the output c_,

    t(Δ) = 2i√(κ_R)[e^iϕλ√(γ)-Δ√(κ_L)]/λ^2+κ_Rγ-Δ(Δ+δ)+i[γ(Δ+δ)+κΔ-2λ√(κ_Lγ)cosϕ].


In the transfer matrix formalism <cit.>, each mirror can be characterized using its polarizability ζ_i, i = R,L.
For the right mirror, the amplitude transmission and reflection coefficients are given by t_R = 1/(1-iζ_R), r_R = iζ_R/(1-iζ_R) and the associated transfer matrix by

    M_R = ([ 1+iζ_R   iζ_R;  -iζ_R 1-iζ_R ]).

An analogous expression holds also for the left mirror with general frequency-dependent polarizability ζ_L(Δ).
Finally, the transfer matrix for free propagation between the two mirrors is given by M_f = (e^iθ,e^-iθ),
where θ = ω/2Γ and Γ = c/2L is the free spectral range.

The transfer matrix of the cavity is obtained by multiplying all three matrices,

    M(Δ)    = M_R M_f M_L 
       = ([    (1+iζ_R)[1+iζ_L(Δ)]e^iθ+ζ_Rζ_L(Δ)e^-iθ  i[1-iζ_L(Δ)]ζ_Re^-iθ+iζ_L(Δ)(1+iζ_R)e^iθ; -i[1+iζ_L(Δ)]ζ_Re^iθ-iζ_L(Δ)(1-iζ_R)e^-iθ    (1-iζ_R)[1-iζ_L(Δ)]e^-iθ+ζ_Rζ_L(Δ)e^iθ ]).

The cavity transmission coefficient can now be obtained as t̃(Δ) = 1/m_22(Δ) which yields

    t̃(Δ)
        = 1/(1-iζ_R)[1-iζ_L(Δ)]e^-iθ+ζ_Rζ_L(Δ)e^iθ.

In going from the general transfer matrix, Eq. (<ref>), to Eq. (<ref>), we expressed the exponentials e^± iθ in terms of the detuning Δ = ω_d-ω instead of the frequency ω.
This step is possible when one notes that the cavity field fulfills the resonance condition

    2ω_dL/c = arctan1/ζ_R+arctan1/ζ_0.


For the example of photonic crystal membranes, we have the polarizability <cit.>

    ζ_L(Δ) = ζ_0γ-2s'Δ/ζ_0/γ+2s'ζ_0Δ.

Here, the parameter s' = ± 1 determines the orientation of the Fano resonance in the mirror's reflectivity.
We also suppose that the resonance frequencies of the cavity mode and the polarizability ζ_L(Δ) coincide.
In this regime, the transmission is maximized for Δ = 0 and reaches unity for ζ_0 = ζ_R.
Note that this condition is different from the coupled-mode description where we assume a general detuning δ between the two.
This discrepancy stems from the different polarizabilities used to determine the cavity resonance frequency:
For the coupled-mode model, the relevant polarizability of the left mirror is 1/ζ_0 whereas it is ζ_0 for the transfer matrix calculation.
We also identified the spectral width of the polarizability (<ref>) with the linewidth of the mirror mode d;
this result follows from coupled-mode analysis of the membrane in the absence of a second mirror <cit.>.
We start the identification by making several observations:

(i)
When Δ = -s'γ/2ζ_0 in the transfer matrix calculation, the polarizability diverges, ζ_L(Δ)→∞;
the reflectivity reaches unity and there is no transmission.
To achieve the same effect in the coupled-mode theory, we must have ϕ = kπ, k∈ℤ as only then can the numerator in Eq. (<ref>) vanish.
The coupling rate

    λ = -s/2ζ_0√(κ_Lγ)

with s = e^iϕ = ± 1 then ensures zero transmission for the same detuning.

(ii)
When the width of the reflectivity is larger than the free spectral range, γ≫Γ, the polarizability ζ_L(Δ)≃ζ_0 for all frequencies of interest.
We then recover the Lorentzian transmission

    t̃(Δ) = κ̃e^iφ/κ̃+iΔ

with the linewidth κ̃= Γ/ζ_0^2 and a polarizability-dependent phase φ; for ζ_0≫ 1, one finds e^iφ≃-i.
The transmission derived from the coupled mode theory in the limit γ→∞ becomes

    t(Δ) = -i√(κ_Rκ_L)/ζ_0/κ_R+κ_L/4ζ_0^2 + i(Δ+δ+sκ_L/ζ_0).

From the real part of the denominator, we must have κ̃= κ_R+κ_L/4ζ_0^2;
moreover, the cavity decays through both mirrors equally if κ_R = Γ/2ζ_0^2 and κ_L = 2Γ.
With these values, the numerator becomes -2iκ_R.
Finally, from the imaginary part of the denominator, we get δ = -sκ_L/ζ_0 = -2s√(κ_Rκ_L).

(iii)
For very narrow polarizability, γ≪Γ, the intensity transmission can be approximated as <cit.>

    |t̃(Δ)|^2 ≃(1 + (1+ζ_0^2)^24Δ^2/(γ+2s'ζ_0Δ)^2)^-1;

it has an asymmetric profile with linewidth γ/2ζ_0^2.
The coupled mode approach in the same limit (i.e., with γ being the smallest rate in the system) yields

    |t(Δ)|^2 ≃(1 + κ_L^2Δ^2/4(κ_Rγ+sΔ√(κ_Rκ_L))^2)^-1.

The transmission spectra in Eqs. (<ref>) and (<ref>) are approximately equal provided ζ_0≫ 1 and s = s';
the former condition is necessary in order to have a well-defined cavity mode so it does not restrict the validity of our approach.

We derived the above results under the assumption that the resonant polarizability of the left mirror is equal to the polarizability of the right mirror, ζ_0 = ζ_R.
Our results can, however, be easily generalized to situations where this is not the case:
The decay rate associated with the right mirror is given by κ_R = Γ/2ζ_R^2 as one would expect and κ_L = 2Γ remains unchanged.
The coupling of the mirror and cavity modes λ, as well as the detuning of the cavity δ, can be expressed in terms of the left-mirror polarizability ζ_0 as

    λ = -s√(κ_0γ),    δ = -2s√(κ_0κ_L).

With these modifications, one can also recover the dynamics of a one-sided cavity, for which ζ_R→∞ (or κ_R→ 0).



We plot the results of the transfer matrix calculation in Fig. <ref>(b) for various values of the mirror mode linewidth.
In the limit γ≪Γ, we obtain the asymmetric lineshape of width κ_ = γ/2ζ_0^2 and, for γ→∞, we obtain the Lorentzian profile of width Γ/ζ_0^2; we discuss these results in the main text.
In the intermediate regime, γ∼Γ, the transmission profile becomes much broader which cannot be replicated in the coupled-mode model with a single cavity mode [see Fig. <ref>(c)].
This discrepancy stems from coupling of the mirror mode to the neighboring cavity modes at frequencies ω_a± = ω_a±Γ [as shown in Fig. <ref>(a)].
When the mirror mode linewidth is comparable to the free spectral range, the mirror mode can mediate interactions between different cavity modes that result in an additional decay of the cavity mode a, leading to a broadened transmission spectrum.
This effect can be included in the coupled-mode model by adding the modes a_± and deriving their coupling to the mirror mode.
This regime is, however, not particularly interesting for applications.








§ OPTOMECHANICAL SIDEBAND COOLING


To evaluate the spectral density of the Langevin force F = g(a+a^†) responsible for cooling, S_F(ω), we solve the equations of motion for the cavity and mirror modes perturbatively, in that we neglect the backaction of the mechanical oscillator on the two modes.
We thus have the Langevin equations


    ȧ   = -(iΔ_a+κ)a -𝒢 d + √(2κ_L)b_ + √(2κ_R)c_,
    ḋ   = -(iΔ_d+γ)d -𝒢 a + √(2γ)b_;


the solution for the cavity mode in the frequency space can be written as

    a(ω) = χ̃_a(ω)[(√(2κ_L)-𝒢χ_d(ω)√(2γ))b_(ω) + √(2κ_R)c_(ω)].

Here, we introduced the bare susceptibilities χ_a^-1(ω) = κ-i(ω-Δ_a), χ_d^-1(ω) = γ-i(ω-Δ_d) and the dressed susceptibility χ̃_a^-1(ω) = χ_a^-1(ω) - 𝒢^2χ_d(ω).
With this solution and the correlation functions for the input fields b_, c_, the spectral density becomes

    S_F(ω)    = 2g^2|χ̃_a(ω)|^2 [κ_R + |√(κ_L)-𝒢χ_d(ω)√(γ)|^2]
       = 2g^2κ_R[γ^2+(ω-Δ_d)^2]+2g^2[γ√(κ_0)-s√(κ_L)(ω-Δ_d)]^2/[(κ+γ)(ω-Δ_d)]^2 + [γ(κ_0+κ_R)-(ω-Δ_d)(ω-Δ_d+2s√(κ_0κ_L))]^2.


To find the final mechanical occupation numerically, we solve the Lyapunov equation for the covariance matrix of the system.
As we describe in detail elsewhere <cit.>, one proceeds by collecting the quadrature operators in the vector r = (X_a,Y_a,X_d,Y_d,q,p)^T,
where X_a = (a+a^†)/√(2), Y_a = -i(a-a^†)/√(2) and similar for d.
Since the dynamics of the system are linear, the time evolution can be fully described by an equation of motion for the covariance matrix,

    V̇ = AV+VA^T+N,

where the matrices A,N depend on the form of the Hamiltonian and the Lindblad terms;
for our system, we have


    A    = ([        -κ       Δ_a  -√(κ_Lγ) -s√(κ_0γ)         0         0;      -Δ_a        -κ  s√(κ_0γ)  -√(κ_Lγ)       -2g         0;  -√(κ_Lγ) -s√(κ_0γ)        -γ       Δ_d         0         0;  s√(κ_0γ)  -√(κ_Lγ)      -Δ_d        -γ         0         0;         0         0         0         0      -γ_m       ω_m;       -2g         0         0         0      -ω_m      -γ_m ]),
    
        N    = ([        2κ         0  2√(κ_Lγ)         0         0         0;         0        2κ         0  2√(κ_Lγ)         0         0;  2√(κ_Lγ)         0        2γ         0         0         0;         0  2√(κ_Lγ)         0        2γ         0         0;         0         0         0         0 2γ_m(2+1)         0;         0         0         0         0         0 2γ_m(2+1) ]);


here, we introduced the initial mechanical occupation at a temperature T, ≃ k_BT/ħω_m.
We find the steady state covariance matrix V̅ (which is possible provided all eigenvalues of A have negative real parts) and determine the final mechanical occupation from the variance of mechanical position and momentum, n_f = (V̅_55+V̅_66-2)/4.





