Dissipative Coupling Coefficients

• Dissipative coupling coefficients are defined as parameters quantifying irreversible losses and decoherence in coupled quantum and classical modes.
• They are computed using a quasi-boson formalism that renormalizes cavity frequencies, resulting in an effective non-Hermitian Hamiltonian for transport analysis.
• The quality metric ζ links dissipative effects to device performance, informing the design of robust quantum networks such as photonic lattices and superconducting qubits.

Dissipative coupling coefficients parameterize the irreversible, non-Hermitian interactions in systems of coupled quantum or classical modes where loss, decoherence, or environmental backaction fundamentally alters the system’s spectrum and dynamics. Across a range of physical platforms—from coupled cavity arrays and optomechanical devices to open photonic lattices and dissipative quantum field theory—the careful characterization and computation of dissipative coupling coefficients provides essential quantitative control over transport, stability, coherence, and nontrivial collective behaviors in open quantum systems.

1. Quasi-Boson Representation and Mode Renormalization

Dissipative coupling emerges most transparently in the "quasi-boson" formalism developed for coupled cavity arrays (CCAs), where the localized cavity mode’s interaction with an infinite bath of harmonic oscillators is formally integrated out (Liu et al., 2010). The effective single-mode Hamiltonian,

$$H = \omega_c c_0^\dagger c_0 + \int d\omega_r\, r^\dagger r + \int d\omega_r\, [\eta^*(\omega_r) r^\dagger c_0 + \text{h.c.}],$$

yields, after diagonalization and the elimination of bath coordinates, a renormalized cavity frequency and an decay term: $$(\omega_c + \delta\omega_c - i\gamma) e_c = \omega\, e_c,$$ where

$$\delta\omega_c = \mathrm{P} \int d\omega_r\, \rho(\omega_r) \frac{|\eta(\omega_r)|^2}{\omega_c - \omega_r}, \qquad \gamma = \pi\rho(\omega_c) |\eta(\omega_c)|^2.$$

The dissipative coupling coefficient, $\gamma$, encapsulates all bath-induced loss and appears as the imaginary component of an effective frequency. This generalizes to CCAs: each site acquires a complex frequency $\omega_\text{eff} = \omega_c - i\gamma$, while the intersite coherent coupling, $\alpha$, is only weakly modified (i.e., $\alpha' \simeq \alpha$), validating the effective non-Hermitian tight-binding picture.

2. Transport, Collective Dynamics, and Scaling Laws

Within this framework, the transport properties of CCAs are governed by the interplay of hopping (set by $\alpha$) and local loss (set by $\gamma$) (Liu et al., 2010). The stationary single-photon transmission amplitude is derived using an ansatz for the wavefunction,

$|\psi\rangle = \sum_{j} e_j b_j^\dagger |0\rangle,$

yielding transmission and reflection coefficients ultimately dependent on $\gamma$ (dissipative coupling), $\alpha$ (coherent coupling), and $N$ (array length).

On resonance, the maximal transmission rate is approximated by

$$T_{\max} \approx \frac{1}{(1 + N\gamma/\xi)^2}, \qquad \xi = 2\alpha\omega_c.$$

This motivates defining a quality scaling parameter,

$$\zeta = \frac{\alpha Q}{N}, \qquad Q = 2\omega_c/\gamma,$$

with $T_{\max} \approx 1/(1+\zeta)^2$, directly relating the dissipative coupling (through $Q$ and $\gamma$) to the scaling of quantum transport.

3. Elimination of Environmental Degrees of Freedom

The primary conceptual and computational advantage of the effective quasi-boson approach is the abstraction of reservoir degrees of freedom. Once $\gamma$ is computed from the microscopic system-bath coupling, all subsequent calculations employ the reduced non-Hermitian Hamiltonian: $$H_\text{eff} = \omega_\text{eff} \sum_j b_j^\dagger b_j - \alpha \omega_\text{eff} \sum_{\langle j,j'\rangle} b_j^\dagger b_{j'}.$$ This approach transforms open-system dynamics into a form amenable to the standard methods of tight-binding physics and scattering theory while rigorously capturing dissipative effects. The only artifact is a correction to bosonic commutation relations of order $1/Q$, negligible in high-$Q$ cavities.

4. Dissipative Coupling Versus Coherent Coupling

Dissipative coupling coefficients (imaginary spectral shifts) are distinguished from coherent (Hermitian) coupling by their fundamental impact on system evolution and spectral signatures. While coherent coupling leads to level repulsion (mode splitting), dissipative coupling may induce level attraction, supermode-mixing, and nonreciprocal transport. In the specific CCA context, increasing $\gamma$ (at fixed $\alpha$) suppresses photon transport, broadens resonances, and defines the transition from underdamped (transport-dominated) to overdamped (localization-dominated) regimes.

5. Generalization to Quantum Networks and Practical Relevance

The effective treatment of dissipative coupling coefficients extends naturally to broad classes of engineered quantum systems, including superconducting qubits, photonic crystals, and topological lattices. The quality parameter $\zeta$ provides a direct quantitative metric for network robustness under loss, with high-$\zeta$ arrays achieving near-unit transmission, while low-$\zeta$ systems suffer rapid decay. The model supports device design exploration, enabling the engineering of environmental couplings for optimal quantum transport, state preservation, and system scalability.

6. Computational and Analytical Techniques

The explicit integration over reservoir frequencies in the complex plane, yielding $\delta\omega_c$ and $\gamma$, is a powerful analytical tool. The one-dimensional structure of the coupled equations allows for efficient recursive and transfer matrix solutions for large arrays, with dissipation included in boundary conditions and matrix elements. Corrections to intersite coupling $\alpha$ by the bath are shown to be minimal, justifying the transfer of all dissipation into on-site complex energy terms.

7. Summary Table

Parameter	Physical Meaning	Analytical Expression
$\gamma$	Dissipative coupling coefficient (local loss rate)	$\pi\rho(\omega_c)|\eta(\omega_c)|^2$
$\omega_\text{eff}$	Complex eigenfrequency	$\omega_c - i\gamma$
$\alpha$	Coherent (overlap) coupling	Overlap integral of neighboring cavity modes
$\zeta$	Quality scaling parameter	$\frac{\alpha Q}{N}$, $Q = 2\omega_c/\gamma$
$T_{\max}$	Maximal transmission rate	$1/(1+N\gamma/\xi)^2$; $\xi = 2\alpha\omega_c$

Conclusion

Dissipative coupling coefficients, exemplified by the local mode decay rate $\gamma$, provide a rigorous and tractable parameterization of environmental loss in coupled photonic arrays. Their integration via the quasi-boson approach enables predictive modeling of quantum transport, identifies scalings governing device performance, and clarifies the dominant physical mechanisms limiting coherence and transmission. This structure underlies the analysis of a broad class of open quantum systems where loss, decoherence, and environmental feedback are intrinsic to operational function or unavoidable due to experimental constraints.