title: Long-distance heat transfer between molecular systems through a hybrid plasmonic-photonic nanoresonator
authors: S. Mahmoud Ashrafi R. Malekfar A. R. Bahrampour Johannes Feist
text: 
  
  
  
  
  Department of Physics, Tarbiat Modares University, Tehran, Iran
  
  Department of Physics, Tarbiat Modares University, Tehran, Iran
  
  Department of Physics, Sharif University of Technology, Tehran, Iran
  
  johannes.feist@uam.es
  Departamento de Física Teórica de la Materia Condensada and
  Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid,
  E-28049 Madrid, Spain
  
  
  We introduce a hybrid plasmonic-photonic cavity setup that can be used to induce
  and control long-distance heat transfer between molecular systems through
  optomechanical interactions. The structure consists of two separated plasmonic
  nanoantennas coupled to a dielectric cavity. The hybrid modes of this resonator
  can combine the large optomechanical coupling of the sub-wavelength plasmonic
  modes with the large quality factor and delocalized character of the cavity mode
  that extends over a large distance (∼μm). We show that this can lead to
  effective long-range heat transport between molecular vibrations that can be
  actively controlled through an external driving laser.
  
  
  Long-distance heat transfer between molecular systems through a hybrid plasmonic-photonic nanoresonator
      Johannes Feist
      December 30, 2023
  =======================================================================================================
  
  
  
  
  
  § INTRODUCTION
  
  Energy transfer between quantum emitters (quantum dots, molecules, atoms,
  …) is a process of fundamental importance for a large range of phenomena in
  quantum information, quantum thermodynamics, quantum biology, photosynthesis,
  solar cells, etc. <cit.>. One powerful strategy to
  modify these processes is by coupling the emitters with an electromagnetic mode
  and mediating transport through photon absorption and
  emission <cit.>. Since the
  light-matter coupling strength is inversely proportional to the effective mode
  volume, the largest single-emitter coupling strengths up to now have been
  reached with nanoplasmonic resonators due to their ability to confine light in
  ultra-small sub-wavelength volumes down to V_p≈ 10^-8λ^3 <cit.>, where λ is the
  free-space wavelength. These capabilities have led to a large range of
  applications of nanoplasmonic systems <cit.>, including
  in the context of quantum electrodynamics (QED) and quantum optics, such as for
  strong coupling with a single molecule at room
  temperature <cit.> or single-photon
  sources <cit.>.
  
  Over the last few years, it has also been realized that the process of
  surface-enhanced Raman scattering, which is a well-known strategy for enhancing
  the Raman scattering signal of molecules by many orders of
  magnitude <cit.>, can alternatively be interpreted
  through the framework of quantum optomechanics <cit.>. In this context, we have recently shown that a localized
  surface plasmon resonance (LSPR) modes can mediate heat transfer between two
  molecules through its optomechanical coupling to the vibrations in each
  molecule <cit.>. However, this approach is limited in practice due
  to the required small (nm-scale) distance between molecules coupled to the same
  sub-wavelength LSPR mode, and additionally suffers from the low quality factors
  (Q ∼ 10) of plasmonic resonators that are unavoidable due to the large
  intrinsic losses of the metals providing the sub-wavelength confinement. Hybrid
  plasmonic-photonic (i.e., metallo-dielectric) cavities present an intriguing
  possibility to circumvent these limitations by combining the strong light-matter
  interactions of plasmonic systems with the possible large lifetimes (high
  quality factors) of dielectric structures. Such approaches have been shown to
  improve the performance of existing applications and allow novel applications
  for a broad range of examples such as strong light-matter
  coupling <cit.>, optical
  trapping <cit.>, surface-enhanced Raman
  scattering <cit.>, label-free detection of
  molecules <cit.>, biosensing <cit.>,
  optoplasmonic sensors <cit.>, or refractometers and nanoparticle
  trapping <cit.>.
  
  
  
  In this article, we show that a hybrid photonic-plasmonic cavity can enable
  efficient long-range heat transfer between spatially separated molecules through
  optomechanical interactions. The setup we propose, sketched in <ref>,
  consists of two metallic nanoantennas coupled to a photonic cavity such as a
  photonic crystal (PC) beam or a dielectric mirror cavity. Such a system supports
  two different kinds of electromagnetic modes, strongly localized LSPR modes on
  each nanoantenna, and an extended cavity mode, which we assume to be driven by
  an external laser. The maximum distance between the nanoantennas is determined
  by the extension of the cavity mode and can be on the order of (several times)
  the laser wavelength. A molecule is placed in the hot spot of each nanoantenna,
  with each molecule coupled to a local heat bath at temperature T_1 (cold) and
  T_2 (hot), respectively. As we will show below, such a system can be used to
  efficiently transfer heat between the molecular vibrational modes, with the
  driving laser providing active control over the heat transfer process.
  
  The paper is organized as follows: in section II, we introduce the quantum
  optomechanical model we use, both in terms of the original uncoupled (LSPR and
  PC) modes as well as in terms of hybrid modes obtained from their coupling. In
  section III, we present the main results, including a discussion of the
  influence of the main parameters of the system, and in particular show the
  consequence of coupling between vibrational modes of different frequencies. We
  conclude with a section summarizing and discussing the results.
  
  
  
  § THEORETICAL MODEL AND FRAMEWORK
  
  
  
  
  Our model, schematically depicted in <ref>, treats three photonic
  modes and two molecules. We note that while the direct quantization of hybrid
  cavity modes is far from trivial <cit.>, we have recently
  shown that they can be represented accurately within a description containing a
  few interacting modes (without counterrotating terms) that each decay
  independently <cit.>. The model we use in the following is based on
  such a description. The three photonic modes are given by one cavity mode, with
  frequency ω_c and bosonic annihilation operator a_c, and two LSPRs,
  with frequencies ω_p,i and bosonic annihilation operators a_p,i (i
  = 1,2). One molecule is assumed to be placed in the hot spot of each LSPR, with
  each molecule represented by a single vibrational mode. These modes are
  approximated as harmonic oscillators, with frequencies ν_j and
  annihilation operators b_j (j = 1,2). We assume that an external laser with
  frequency ω_l and amplitude Ω drives the cavity mode. The total
  Hamiltonian of the system in the rotating frame of the laser and within the
  rotating wave approximation can then be written as (here and in the following,
  we set ħ=1)
  
  
      H    = H_ph + H_m + H_I + H_d,
      
        H_ph   = ∑_αδ_α a_α^† a_α + ∑_i=1^2 g_cp (a_c^† a_p,i + a_c a_p,i^†),
      
        H_m   = ∑_j=1^2 ν_j b_j^† b_j,
      
        H_I   = -∑_α,j g_α j a_α^† a_α (b_j^† + b_j)
      
        H_d   = -iΩ (a_c^† - a_c)
  
  
  where δ_α = ω_α - ω_l, the sum over α includes
  the three photonic modes and g_cp accounts for the interaction between the
  cavity mode and the LSPRs, which depends on the placement of the nanoantennas
  relative to the optical cavity <cit.>. Here and in the
  following, we consider two identical plasmonic cavities placed at equivalent
  positions sufficiently separated from each other that direct plasmon-plasmon
  interactions are negligible. The interaction between photonic modes and
  vibrations, H_I is approximated through their optomechanical
  coupling, which is justified when ω_α≫ν_j <cit.>. The optomechanical
  coupling constants are given by g_α j = ω_α Q_j^0
  R_j/2εε_0V_α, where Q_j^0 is the zero
  point amplitude of vibration j, R_j is its Raman activity,
  ε is the relative permittivity of the surrounding medium and
  V_α is the effective mode volume of the photonic mode. Note that the
  “bare” photonic mode volume is large (V ∼λ^3 ∼ 10^9nm^3), so that its
  direct optomechanical coupling to the molecules is negligible, and that each
  plasmonic antenna only interacts with its “own” molecule, g_p_1 2 =
  g_p_2 1 = 0.
  
  In addition to the coherent dynamics described by the Hamiltonian, we also model
  the incoherent dynamics due to the interaction between the different components
  of the system and their environment. The dynamics of the system density matrix
  is described within the Lindblad master equation formalism <cit.>:
  
      dρ/dt = -i [H,ρ] + ∑_α L_a_α[ρ] + ∑_j L_b_j[ρ],
  
  with 
  
      L_a_α[ρ]    = κ_α D_a_α[ρ],
      
        L_b_j[ρ]    = γ_j (n̅_j + 1) D_b_j[ρ] + γ_jn̅_j D_b_j^†[ρ].
  
  Here, κ_α is the decay rate of the photonic mode α,
  γ_j is the molecular damping rate of molecule j, and n̅_j =
  1/[exp(ν_j/k_B T_j) - 1] is the mean phonon
  number of molecule j when it is in thermal equilibrium with its associated
  bath at temperature T_j. Finally, D_A[ρ] is a Lindblad dissipator,
  
      D_A[ρ] = A ρ A^† - 1/2{ A^†A, ρ}.
   
  
  In order to quantify the heat transfer between the molecules, we follow the same
  approach as in our previous paper <cit.>, which we summarize in the
  following: We first solve for the steady state of the system,
  dρ_ss/dt = 0, and then extract an
  effective temperature for the molecular vibrations,
  
      T_j^eff = ν_j/k_Blog(1 + 1/n_j),
  
  based on the average phonon number n_j = ⟨ b_j^† b_j⟩
  = (b_j^† b_jρ_ss). For this effective temperature
  to correspond to a physical temperature, the population of the energy levels
  should again follow a thermal distribution. We have checked for all the results
  presented below that this is indeed the case, i.e., that the steady-state
  distributions of the phonon populations are well approximated by thermal
  Boltzmann distributions, and the effective temperatures obtained can thus indeed
  be interpreted as the steady-state physical temperatures of the respective
  vibrational modes.
  
  All the numerical results have been obtained using the QuTiP
  package <cit.>, and plots have been prepared using
  matplotlib <cit.>. We note that in contrast to our
  previous work <cit.>, which used only a single photonic mode, we
  here treat a relatively big quantum system consisting of five bosonic modes
  (three photonic and two vibrational) within a density matrix description, such
  that the numerical size of the density matrix is 5^N × 5^N if N states
  are used for each mode. The size of the (numerically sparse) Liouvillian
  superoperator is the square of that number. In order to obtain numerical
  convergence while keeping the size of the calculations manageable, we use a
  basis in which the total number of excitations (photons + phonons) is
  restricted, instead of using a cutoff for each mode separately. The results
  presented below use a maximum number of five excitations, which we have checked
  to give converged results for the parameters used.
  
  
  
  § RESULTS AND DISCUSSION
  
  
  
  
  We now investigate long-range energy transfer between two molecules, each
  coupled to a plasmonic nanoantenna with a low-Q resonance, with the two antennas
  coupled to the same high-Q cavity mode. The model we use is similar to the one
  proposed by Kamandar Dezfouli and Hughes <cit.>, with the
  addition of a second plasmonic antenna and molecule. It consists of a photonic
  crystal supporting a mode with frequency ω_c = 1.61 eV and a high
  quality factor, Q_c = 3 × 10^5 (i.e. κ_c = 5.4 ×
  10^-2 meV), and two identical bowtie nanoantennas separated by a relatively
  large distance (∼μm), with a dipolar LSPR at frequency ω_p,1 =
  ω_p,2 = 2.2 eV, and relatively low quality factor of Q_p,1 = Q_p,2
  = 40 (i.e., κ_p,1 = κ_p,2 = 55 meV). One molecule is placed in
  the hot spot (i.e., the center) of each nanoantenna. We first treat the case of
  two identical molecules, with a vibrational mode with angular frequency ν_1
  = ν_2 = 40 meV, and vibrational damping rate of γ_1 = γ_2 =
  0.1 meV. Each molecule is coupled to a local heat bath at a different
  temperature, where here and in the following we set T_1 = 77 K, T_2 =
  300 K. The optomechanical coupling rate for the LSPR modes is taken as g_p =
  75 meV, corresponding to close-to-resonant Raman
  transitions <cit.>, while the direct optomechanical
  coupling to the cavity mode is so small as to have negligible effect (due to the
  large mode volume). However, the coupling between the cavity mode and each LSPR,
  with coupling constant g_cp = 200 meV, mediates an indirect coupling between
  the LSPR modes, and thus between the molecules.
  
  
  
  We focus on the case where the external laser drives the cavity mode with red
  detuning, ω_l = 1.41 eV. <ref> shows the effective
  temperature of each molecule as a function of the external laser driving
  strength Ω. As expected, for Ω=0, the photonic modes have no
  influence on the molecules, and each molecular vibration is in equilibrium with
  its local heat bath. When the laser is turned on, there are two effects on the
  molecular system. First, the optomechanical coupling for each molecule in
  isolation leads to heating/cooling of the vibrations through Stokes/anti-Stokes
  transitions, such that they are driven out of equilibrium with their respective
  heat baths. This well-known single-molecule effect <cit.> is shown
  in dashed lines in <ref>. For the parameters considered here, this
  results in heating for both molecules, significant for the colder molecule
  (T_1^eff increases from 77 K to ≈170 K), and less
  pronounced for the hotter molecule (T_2^eff increases from 300 K
  to ≈315 K). Second, there is an effective molecule-molecule coupling
  mediated by the photonic modes (through the successive couplings
  molecule-LSPR-cavity-LSPR-molecule), which enables energy transfer between the
  molecules, the efficiency of which can be estimated from the effective molecular
  temperatures (solid lines in <ref>) compared to the single-molecule
  cases. As seen in <ref>, this can be a significant effect, enabling
  efficient transfer of energy between the molecules under external pumping of the
  high-Q cavity mode. For example, for Ω = 35 meV, we obtain
  T_1^eff≈ 220 K, T_2^eff≈ 280 K, showing
  that the hot molecule is efficiently cooled through long-distance heat transfer
  to the cooler molecule, which is significantly heated in return. The proposed
  setup could thus indeed be used for externally controlling the heat or energy
  transfer between molecules that are spatially separated by significant
  distances. The pumping rate is the main parameter controlling this effect, and
  increasing it improves the energy exchange, causing the effective temperatures
  to approach each other.
  
  Additional insight into the long-range heat transfer can be obtained by
  considering a simplified model where the three photonic modes are diagonalized
  to obtain hybrid modes (see sec:hybrid_framework for details). In this
  case, one obtains one high-Q and two lower-Q hybrid modes (one of which is a
  “dark” plasmonic mode consisting of the antisymmetric combination of plasmonic
  antenna modes and does not hybridize with the cavity-like mode). The high-Q mode
  is dominated by the cavity mode contribution. We thus label it as the
  “cavity-like” mode in the following. Considering only the optomechanical
  interaction with the cavity-like mode (dots in <ref>) reproduces the
  main effect of optomechanically induced heat transfer, although it
  underestimates the laser-induced heating, in particular of the hotter molecule.
  Adiabatically eliminating the cavity-like mode within this single-mode
  approximation then gives an effective molecule-molecule coupling term Λ
  (b_1^† b_2 + b_1 b_2^†) <cit.>. The effective
  temperatures obtained within this approximation (dashed black lines in
  <ref>, with the corresponding value of Λ shown in the upper
  axis) reproduce the single-mode results almost perfectly.
  
  We next treat the influence of other parameters on long-distance heat flow
  between the molecules. We show the effective temperature of the molecules as a
  function of the coupling rate γ = γ_1 = γ_2 between each
  vibration and its local heat bath in <ref>(a). This rate determines
  the rate of heat exchange with the local bath, and the temperatures of the
  vibrational modes thus approach the bath temperatures as γ is increased.
  For the “hot” molecule 2, this is enough to make the temperature of the
  molecule approach its undriven value (T_2 = 300 K). While the effect is also
  very noticeable for the “cold” molecule, with a decrease in temperature from
  ≈220 K to ≈110 K as γ is increased from 0.1 meV to
  1 meV, it is not enough to completely counteract the externally induced heat
  transport. Again, the single-mode approximation only taking into account the
  cavity-like hybrid mode reproduces the observed trends quite well while
  underestimating the laser-induced heating of the hotter molecule.
  
  We additionally investigate the effect of the photonic system parameters on the
  heat transfer effect. In particular, as shown in <ref>(b), an increase
  in the plasmonic cavity loss κ_p = κ_p,1 = κ_p,2 leads to
  a reduction of heat transfer, while simultaneously inducing more heating on the
  single-molecule level. As the single-mode approximation that only takes into
  account the high-Q cavity-like mode underestimates the plasmon-induced heating,
  it is not capable of reproducing this trend accurately, and the effective
  temperatures within this approximation are approximately constant as a function
  of κ_p, see the dots in <ref>(b). The effect seen here shows
  that it is desireable to use plasmonic nanoantennas of the highest quality
  possible to minimize direct optomechanical heating while simultaneously reaching
  large effective optomechanical coupling strengths. Similarly, we find that
  improving the quality factor of the delocalized cavity mode (i.e., decreasing
  its loss rate) also leads to improved heat transfer (not shown).
  
  
  
  
  
  We now treat the non-symmetric case where the vibrational frequencies of the two
  molecules are not identical (while keeping all other parameters fixed for
  simplicity), i.e., we explore whether it is possible to enable heat transfer
  between molecules even if their vibrational frequencies are not equal. In
  <ref>, we plot the effective vibrational temperatures of the two
  molecules when the vibrational frequency ν_2 of the “hot” molecule is
  changed, with ν_1 fixed at 40 meV and all other parameters as in
  <ref>. We find relatively efficient heat transfer only when the two
  vibrations are close to resonance (ν_1≈ν_2). These results are
  consistent with the case where two molecules are coupled to the same
  plasmonic nanocavity mode <cit.>, and imply that a
  single-vibrational-mode approximation is justified.
  
  To gain a better understanding of this effect, we next analyze the power
  spectral density (PSD) of the involved modes, which measures their oscillation
  spectrum as a function of frequency <cit.>. The PSD for a mode with
  annihilation operator A is given by the Fourier transform of the correlation
  spectrum of the operator in the steady state, 
  
      S_A(ω) = ∫ e^- iω t⟨ A^†(t)A(0) ⟩_ssdω.
  
  The PSD for the symmetric case of identical molecular vibrations is shown in
  <ref>(a). For the photonic mode basis, we here use the hybrid photonic
  modes obtained after diagonalizing the coupled cavity and LSPR modes. This
  figure shows that the cavity-like hybrid mode (black dashed line) is the
  principal photonic mode involved in the dynamics, with a modulation imprinted on
  it at frequencies close to the vibrational ones. As discussed previously, this
  mode mediates the effective long-range molecule–molecule coupling. In the
  current regime, the effective linewidth of the PSD of the cavity-like mode is
  not determined by its loss rate (κ_- ≈ 8 meV), but much
  closer to the molecular vibrational linewidth. When the two molecular vibrations
  are detuned from each other, shown in <ref>(b) for vibrational
  frequencies ν_1 = 40 meV and ν_2 = 37 meV, each molecule only
  oscillates at its native frequency, and the cavity-like mode has two essentially
  independent peaks at ν_1 and ν_2. The off-resonant fluctuations induced
  on the cavity-like mode by the two vibrational modes then prevent effective
  long-range energy transfer between them.
  
  
  
  § SUMMARY & CONCLUSIONS:
  
  
  To summarize, we have investigated the optomechanical heat transfer mechanism
  between two spatially separated molecules at different local temperatures placed
  in a hybrid plasmonic-photonic nanoresonator. The hybrid cavity considered
  consists of two plasmonic nanoantennas supporting LSPR modes that are both
  coupled to the same high-quality cavity mode (such as supported by a photonic
  crystal cavity or a Fabry-Perot cavity with highly reflective mirrors). The
  cavity mode is driven by an external red-detuned laser. We have shown that in
  such a hybrid setup, the cavity-like mode, which itself has negligible
  optomechanical coupling to the molecular vibrations, can behave like a mediator
  that provides an effective molecule–molecule coupling over large distances of
  the order of several free-space wavelengths, while the strongly sub-wavelength
  plasmonic modes provide the required large optomechanical coupling strengths.
  The hybrid setup considered can thus induce long-range heat transfer between
  molecules. Additionally, the heat transfer is fully controlled by an external
  driving laser, and can thus be dynamically turned off or on. Furthermore, we
  have found that this mechanism only occurs for close-to-resonant vibrational
  modes, which can be understood by considering the molecule-induced oscillation
  of the hybrid cavity modes through their power spectral density.
  
  
  This work has been funded by the European Research Council through grant
  ERC-2016-StG-714870 and by the Spanish Ministry for Science, Innovation, and
  Universities – Agencia Estatal de Investigación through grants
  RTI2018-099737-B-I00, PCI2018-093145 (through the QuantERA program of the
  European Commission), and MDM-2014-0377 (through the María de Maeztu program for
  Units of Excellence in R&D), as well as through a Ramón y Cajal grant (JF) and
  support from the Iranian Ministry of Science, Research and Technology (SMA).
  
  
  
  
  
  
  § HYBRID MODES
  
  
  We here discuss the properties of the hybrid modes obtained by diagonalizing the
  “photonic” subsystem consisting of the cavity mode and two localized surface
  plasmon resonances. The photonic Hamiltonian can be written as H_ph
  = A⃗^†ℋA⃗, where A⃗ =
  (a_c,a_p,1,a_p,2)^T collects the photonic annihilation operators and the
  matrix ℋ is
  
      ℋ = [  δ_p    0 g_cp;    0  δ_p g_cp; g_cp g_cp  δ_c ].
  
  Diagonalization of ℋ gives the hybrid mode energies
  
      δ_±   = δ_p + δ_c/2±1/2√((δ_p - δ_c)^2 + 8g_cp^2),
      δ_D   = δ_p,
  
  and annihilation operators
  
      a_-   =  cosθ a_c + sinθ/√(2) (a_p,1 + a_p,2),
      
        a_+   = -sinθ a_c + cosθ/√(2) (a_p,1 + a_p,2),
      
        a_D   = 1/√(2) (a_p,1 - a_p,2),
  
  where tan2θ = √(8) g_cp / (δ_p - δ_c). If the
  coupling g_cp is not too large compared to the detuning δ_p -
  δ_c, the hybrid modes are a `small' rotation of the original basis and
  can be identified as a cavity-like hybrid mode (a_-), an LSPR-like hybrid mode
  (a_+), and an LSPR dark mode a_D that has no cavity contribution. The
  original operators can be represented in the new basis as
  
      a_c   = cosθ a_- - sinθ a_+,
      
        a_p,1   = 1/√(2)(sinθ a_- + cosθ a_+ + a_D),
      
        a_p,2   = 1/√(2)(sinθ a_- + cosθ a_+ - a_D).
  
  The changed parts of the Hamiltonian in the new hybrid basis are given by
  
  
      H_ph   = ∑_βδ_β a_β^† a_β,
      
          H_I   = -∑_β,j g_β a_β^† a_β (b_j^† + b_j) 
          -∑_β'>β,j g_β' β (a_β^† a_β' + a_β a_β'^†) (b_j^† + b_j),
      
          H_d   = i Ωcosθ (a_-^† - a_-) - i Ωsinθ(a_+^† - a_+),
  
  
  where β∈-,+,D runs over the hybrid modes, and
  
  
      g_-   = sin^2θg_p/2,     g_+-   = sinθcosθg_p/2,
      
        g_+   = cos^2θg_p/2,     g_D-   = sinθg_p/2,
      
        g_D   = g_p/2,                    g_D+   = cosθg_p/2.
  
  
  
  
  
  The dynamics is then described by the master equation
  
      dρ/dt = -i [H,ρ] + ∑_β L_a_β[ρ] 
        + ∑_j L_b_j[ρ] + L̅_±[ρ],
  
  where
  
  
      L_a_β[ρ]    = κ_β D_a_β[ρ],
      L̅_±[ρ]    = κ_± (D_a_+,a_-[ρ] + D_a_-,a_+[ρ]),
  
  
  where D_a,b[ρ] = aρ b^† - 1/2{ b^† a, ρ} and
  
  
      κ_D   = κ_p,
      κ_-   = cos^2θκ_c + sin^2θκ_p,
      κ_+   = cos^2θκ_p + sin^2θκ_c,
      κ_±   = (κ_p - κ_c) sinθcosθ.
  
  
  The properties of the hybrid modes are shown schematically in
  <ref>, where each mode is represented by a Lorentzian at
  frequency ω_β = δ_β + ω_l, with linewidth
  κ_β and amplitude g_β. The driving laser is indicates as a
  dotted blue-green line at ω_l = 1.41 eV. For the parameters considered,
  the cavity-like mode a_- thus clearly gives the most significant contribution
  since it is close to resonance with the external laser and has a strong
  optomechanical interaction. The simplified effective Hamiltonian obtained by
  only taking into account the hybrid cavity-like mode is given by
  
      H_hc≈δ_-a_-^†a_- + ω_1b_1^†b_1 + ω_2b_2^†b_2 + iΩcosθ (a_-^† - a_-)
      
        - g_- a_-^†a_- (b_1^† + b_1 + b_2^† + b_2).
  
  The results obtained with this Hamiltonian are shown in the main text as the
  “cavity-like” approximation.
  
  
  
  