Holstein–Tavis–Cummings Hamiltonians

• The Holstein–Tavis–Cummings Hamiltonian is a unified model that combines collective light–matter interactions with local vibrational modes in organic molecules.
• It employs the Tavis–Cummings term for √N scaling of the vacuum-Rabi splitting and the Holstein coupling for capturing Franck–Condon sidebands and dephasing effects.
• The framework supports analytic and numerical investigations into molecule counting, spectral shifts, and quantum control in cavity QED systems.

The Holstein–Tavis–Cummings (HTC) Hamiltonian formalism provides a unified theoretical framework that integrates the effects of collective light–matter coupling—characteristic of the Tavis–Cummings model—with local vibronic modes described by the Holstein model. It is explicitly constructed for systems composed of $N$ identical organic molecules, each possessing one electronic two-level system and one vibrational mode, interacting with a single-mode optical cavity. The HTC Hamiltonian enables analytic and numerical investigations into phenomena including vacuum-Rabi splittings with $\sqrt{N}$ scaling, Franck–Condon sidebands, vibrationally induced dephasing and Lamb shifts, and the modification of molecular conformation in cavity transmission and fluorescence spectra (Zhang et al., 2021).

1. Definition and Structure of the HTC Hamiltonian

The HTC Hamiltonian, in a frame rotating at the driving laser frequency $\omega_l$ ($\hbar\equiv1$), is given by:

$$H_{\text{HTC}} = H_{\text{cav}} + \sum_m \left(H_{\text{mol},m} + H_{\text{int},m}\right) + H_{\text{drive}}$$

where:

• $H_{\text{cav}} = \Delta_c\, a^\dagger a$, $\Delta_c = \omega_c - \omega_l$, and $a$ ($a^\dagger$) are cavity annihilation (creation) operators, with decay rate $\kappa$.
• $$H_{\text{mol},m} = \nu\, b_m^\dagger b_m + [\omega_{00} + \lambda \nu (b_m + b_m^\dagger)]\, \sigma_m^\dagger \sigma_m$$, invoking the Holstein coupling. Here, $b_m$ ($b_m^\dagger$) are vibrational operators, $\nu$ is the vibrational frequency, $\lambda$ is the Huang–Rhys factor, and $\sigma_m = |g\rangle_m \langle e|_m$ is the exciton lowering operator.
• $$H_{\text{int},m} = g\, (a^\dagger \sigma_m + \sigma_m^\dagger a)$$ is the Tavis–Cummings term (rotating-wave approximation, $g \ll \omega_c, \omega_{00}$).
• $$H_{\text{drive}} = i\eta\,(a e^{i\omega_l t} - a^\dagger e^{-i\omega_l t})$$ describes a weak coherent probe ($\eta \ll \kappa$).

The master equation includes Lindblad dissipators for cavity loss ($\kappa\, \mathcal{L}[a]$), electronic spontaneous emission ($\Gamma\, \mathcal{L}[\sigma_m]$), pure dephasing ($$\Gamma_\phi\, \mathcal{L}[\sigma_m^\dagger\sigma_m]$$), and vibrational damping ($\gamma\, \mathcal{L}[b_m]$) (Zhang et al., 2021).

2. Cavity Transmission Spectrum and Quantum Langevin Approach

The quantum Langevin formalism is employed to derive the cavity transmission spectrum. In the Heisenberg–Langevin picture (rotating at $\omega_l$), defining the displaced operator $\tilde{\sigma}_m \equiv D_m^\dagger\,\sigma_m$ with the polaron displacement $D_m = \exp[\lambda(b_m-b_m^\dagger)]$, one obtains: $$\begin{align*} \frac{da}{dt} &= -(\mathrm{i}\Delta_c + \kappa)a - \mathrm{i}g\sum_m \sigma_m + \sqrt{2\kappa_1}A_{\text{in},1} + \sqrt{2\kappa_2}A_{\text{in},2} + \eta\ \frac{d\tilde{\sigma}_m}{dt} &\approx -(\mathrm{i}\Delta+\Gamma_\perp)\tilde{\sigma}_m - \mathrm{i}g D_m^\dagger a + \sqrt{2\Gamma_\perp} D_m^\dagger \sigma_{\text{in},m} \end{align*}$$ with $\Delta = \omega_{00} - \omega_l$, $\Gamma_\perp = \Gamma + 2\Gamma_\phi$.

Assuming weak driving $(\langle \sigma^\dagger \sigma \rangle \ll 1)$, integration and steady-state analysis yields the mean cavity amplitude: $$\langle a \rangle_{\text{ss}} = \frac{\eta}{\mathrm{i}\Delta_c + \kappa + N g^2 \chi}$$ where

$$\chi = \lim_{s\rightarrow0} \sum_{k=0}^{\infty} \frac{(\lambda^2)^k e^{-\lambda^2}}{k!} \frac{1}{\mathrm{i}(\Delta + k\nu) + (\Gamma_\perp + k\gamma)}$$

The cavity transmission is then

$$T(\omega_l) \propto \frac{|2\sqrt{\kappa_1\kappa_2}|^2}{|\mathrm{i} \Delta_c + \kappa + N g^2 \chi|^2}$$

(Zhang et al., 2021).

3. Adiabatic Elimination of Vibrational Modes

In the prevalent regime $\gamma \gg \kappa, \Gamma_\perp$ typical in organic-in-solution systems, vibrational relaxation dominates. The approximation $$\langle a(t_1) D_m(t) D_m^\dagger(t_1) \rangle \approx \langle a(t_1) \rangle \langle D_m(t) D_m^\dagger(t_1) \rangle$$ allows factorization. The vibrational correlation function reduces to: $$\langle D_m(t) D_m^\dagger(t_1) \rangle = \exp\left[ -\lambda^2 (1 - e^{-(\gamma + i\nu)(t-t_1)}) \right]$$ In the $\gamma \rightarrow \infty$ limit, this enforces Markovianity, collapsing the multi-phonon Franck–Condon series to the zero-phonon line with a vibrationally induced shift and additional dephasing.

The effective non-Hermitian Hamiltonian becomes: $$H_{\text{eff}} = \Delta_{\text{eff}}\sum \sigma_m^\dagger\sigma_m + \frac{\mathrm{i}}{2}\Gamma_{\text{eff}} \sum \sigma_m^\dagger\sigma_m + g\sum(a^\dagger \sigma_m + \text{H.c.}) + \Delta_c a^\dagger a - \frac{i\kappa}{2}a^\dagger a$$ with $\Delta_{\text{eff}} = \operatorname{Im}[1/\chi]$ and $\Gamma_{\text{eff}} = \operatorname{Re}[1/\chi]$.

This non-Hermitian Hamiltonian incorporates vibrationally induced Lamb shifts and vibronic dephasing, and captures the Markovian role of vibrational environments in polaritonic spectroscopy (Zhang et al., 2021).

4. Analytic Calculation of Polaritonic States

On resonance ($\Delta_c = \Delta = 0$), the system reduces to an effective coupling between the cavity field $a$ and the "bright" molecular supermode $B = N^{-1/2}\sum_m \sigma_m$. The $2\times2$ non-Hermitian problem

$$H_{2\times2} = \begin{pmatrix} -\frac{\mathrm{i}\kappa}{2} & g\sqrt{N}\ g\sqrt{N} & -\frac{\mathrm{i}\Gamma_{\text{eff}}}{2} \end{pmatrix}$$

yields eigenfrequencies $\omega_\pm - \mathrm{i}\Gamma_\pm$, where: $$\omega_\pm = -\frac{\Delta_{\text{eff}}}{2} \pm \operatorname{Im} \sqrt{N g^2 - \left(\frac{\kappa - \Gamma_{\text{eff}}}{2}\right)^2}$$

$$\Gamma_\pm = \frac{\kappa + \Gamma_{\text{eff}}}{2} \mp \operatorname{Re} \sqrt{N g^2 - \left(\frac{\kappa - \Gamma_{\text{eff}}}{2}\right)^2}$$

Under strong coupling $N g^2 \gg [(\kappa - \Gamma_{\text{eff}})/2]^2$, one recovers the standard polariton splitting $\omega_\pm \approx \pm g\sqrt{N}$. The lower-polariton resonance is thus shifted by

$\delta\omega_{-} = -g\sqrt{N}$

This $\sqrt{N}$ scaling forms the basis for molecule counting or detection of ultracold ensembles via Rabi splitting measurement (Zhang et al., 2021).

5. Physical Implications and Interpretation

The component Hamiltonians represent separate physical mechanisms:

• $H_{\text{cav}}$ describes the lossy, driven single-mode cavity.
• $H_{\text{mol},m}$ (Holstein) captures local vibronic effects (Franck–Condon sidebands, temperature-dependent spectral lineshapes).
• $H_{\text{int},m}$ (Tavis–Cummings) enables collective light–matter coupling, inducing a Rabi splitting $\propto g\sqrt{N}$.
• The rotating-wave approximation applies for $g \ll \omega_c, \omega_{00}$, ensuring the neglect of non-resonant terms.
• Adiabatic elimination introduces an effective Markovian vibrational bath, yielding collective Lamb shifts ($\Delta_{\text{eff}}$) and extra broadening ($\Gamma_{\text{eff}}$).

The table below summarizes the key HTC Hamiltonian constituents and their physical content:

Term	Mathematical Representation	Physical Role
$H_{\text{cav}}$	$\Delta_c\, a^\dagger a$	Cavity photon mode (lossy, driven)
$H_{\text{mol},m}$	$$\nu b_m^\dagger b_m + [\omega_{00} + \lambda \nu (b_m + b_m^\dagger)]\sigma_m^\dagger \sigma_m$$	Local Holstein (vibronic) couplings
$H_{\text{int},m}$	$g(a^\dagger \sigma_m + \sigma_m^\dagger a)$	Collective Tavis–Cummings coupling
$H_{\text{drive}}$	$$i\eta(a e^{i\omega_l t} - a^\dagger e^{-i\omega_l t})$$	Weak probe drive

(Zhang et al., 2021)

6. Applications and Significance

The HTC framework enables several key applications:

• The $\sqrt{N}$ scaling of the polariton splitting enables accurate estimation of the number of coupled molecules, including regimes where direct fluorescence is inaccessible (ultracold or weak emission cases).
• Franck–Condon dephasing and vibronic sidebands, which are encoded within the HTC structure, manifest as changes in transmission and fluorescence spectra as $N$ is varied. This spectral evolution provides a handle on ensemble geometry and molecular configuration.
• The formalism supports the analysis of spectroscopic signatures arising from collective vibronic polaritons, with implications for probing and manipulating chemical reactivity, as well as energy and charge transport phenomena in organic materials.
• Analytic solutions for the transmission and fluorescence spectra, including the effects of Markovian vibrational baths, enable the characterization and control of organic microcavity polaritons at the quantum level (Zhang et al., 2021).

In summary, the Holstein–Tavis–Cummings Hamiltonian offers a framework for capturing the interplay of collective and local vibronic effects in cavity QED systems with organic molecules. The resulting spectroscopy reveals $\sqrt{N}$-scale vacuum–Rabi splittings, vibrational sidebands, and environment-induced shifts and broadenings, providing practical methodologies for molecule counting, conformational probing, and quantum-level chemical control.