Dissipaton Theory in Open Quantum Systems

• Dissipaton theory is a formalism for exact, nonperturbative modeling of open quantum systems, encoding environmental effects as algebraic dissipatons with linear and quadratic couplings.
• It decomposes bath operators into statistically independent dissipatons with exponential memory kernels, enabling a systematic construction of a coupled system–bath hierarchy via generalized Wick’s theorems.
• The DEOM framework derived from this theory accurately simulates non-Gaussian bath phenomena, offering practical insights for optical, transport, and relaxation processes beyond linear response.

Dissipaton theory is a formalism for the exact, nonperturbative, and hierarchy-based description of open quantum system dynamics, in which the environment (bath) exerts both linear and nonlinear (e.g., quadratic) couplings on a system. In this approach, environmental influences are encoded as algebraic “dissipatons”—statistical quasi-particle operators that allow the systematic construction of a coupled system–bath hierarchy. The theory’s algebraic backbone—rooted in generalized diffusion equations and an extended Wick's theorem—enables rigorous treatment of non-Gaussian (nonlinear) environments, going well beyond what linear response or conventional Gaussian assumptions can capture. The dissipaton-equation-of-motion (DEOM) framework translates these algebraic principles into explicit equations of motion for a “ladder” of dissipaton density operators (DDOs), capturing the entanglement and dynamical interplay between quantum systems and complex, possibly strongly nonlinear, baths.

1. Hamiltonian Structure and Need for Beyond-Gaussian Theories

A fundamental aspect of dissipaton theory is the explicit representation of general system–bath interaction Hamiltonians, specifically incorporating both linear and nonlinear coupling terms: $$H_\text{tot} = H_S + h_B + \hat Q ( \alpha_1 \hat x_B + \alpha_2 \hat x_B^2 )$$ where $H_S$ is the system Hamiltonian, $h_B$ the bath Hamiltonian (often a set of harmonic oscillators), $\hat Q$ a system dissipative operator, and $\alpha_1$, $\alpha_2$ respectively quantify the strengths of linear and quadratic coupling.

For $\alpha_2 = 0$, bath statistics are Gaussian: standard approaches—e.g., quantum master equations—apply, relying on the conventional Wick’s theorem. However, when $\alpha_2 \neq 0$, nonlinear bath couplings push the system into the domain of non-Gaussian statistics, precluding complete characterization via linear response and demanding a generalized algebraic framework.

2. Dissipatons—Statistical Quasi-Particles and Bath Decomposition

At the core of dissipaton theory is the representation of the bath operator $\hat x_B$ as a sum over elementary dissipaton operators: $\hat x_B = \sum_k \hat f_k$ Each $\hat f_k$ corresponds to an exponential term in the bath autocorrelation function (FDT expansion): $$\langle \hat x_B(t) \hat x_B(0) \rangle = \sum_k \eta_k e^{-\gamma_k t}$$ These $\hat f_k$ are assigned exponential memory kernels: $$\langle \hat f_k(t) \hat f_j(0) \rangle \propto \delta_{kj}\,\eta_k e^{-\gamma_k t}$$. This allows the replacement of the many-body bath with a set of statistically independent dissipatons, each carrying a single damping rate $\gamma_k$ (generalized diffusion equation).

This dissipaton expansion is essential: it enables the mapping of bath-induced system–bath entanglement onto an algebraic hierarchy and provides the foundation for extending beyond Gaussian statistics.

3. Dissipaton Density Operators, Hierarchy Construction, and Generalized Wick’s Theorem

The dynamics—both of the system and its entanglement with the bath—are entirely encoded in dissipaton density operators (DDOs): $$\rho_n^{(n)}(t) = \operatorname{Tr}_B \{ ( \hat f_K^{n_K} \cdots \hat f_1^{n_1} )^\circ \rho_\text{tot}(t) \}$$ with $n = n_1 + n_2 + \cdots + n_K$, and the “$\circ$” denoting an irreducible (connected) operator product.

To construct the hierarchy, two key algebraic structures are required:

• Generalized Diffusion Equation: The commutator with the bath Hamiltonian induces exponential damping, giving for any $n$-dissipaton operator,

$$i\,\operatorname{Tr}_B \big\{ [ (\cdots)^\circ, h_B ] \rho_\text{tot}(t) \big\} = \left(\sum_k n_k \gamma_k \right) \rho^{(n)}_n(t)$$

• Generalized Wick’s Theorem (GWT): For linear coupling (GWT-1), contraction rules involving one additional dissipaton preserve the hierarchical structure; for nonlinear/quadratic coupling (GWT-2), pairs of dissipatons must be added, inducing a more intricate coupling between DDOs at different hierarchy tiers:

$$\operatorname{Tr}_B \left\{ (\cdots)^\circ ( \hat f_j \hat f_{j'} ) \rho_\text{tot}(t) \right\} = \rho_{jj'}^{(n+2)}(t) + \hat f_j \hat f_{j'} \rho^{(n)}(t) + \cdots$$

(additional contraction/summation terms as detailed in the full GWT-2 expansion).

The closure these theorems provide ensures that the full system–bath dynamics is consistently and exactly encoded in the DDO hierarchy, with all non-Gaussian memory effects included.

4. Extended DEOM Evolution Equation: Linear and Quadratic Couplings

The full evolution equation for DDOs incorporating both linear and quadratic couplings [see Eq. (9) in (Xu et al., 2016)] is: $$\begin{aligned} \frac{d}{dt} \rho_n^{(n)}(t) = & - \left[ i \mathcal{L}_\text{eff} + \sum_k n_k \gamma_k \right] \rho_n^{(n)}(t) - i2\alpha_2 \sum_{k,j} n_k \mathcal{C}_k \rho^{(n)}_{(-k,+j)}(t) \ & - i\alpha_2 \sum_{k,j} \left( \mathcal{A} \rho^{(n+2)}_{k,j}(t) + n_k (n_j - \delta_{kj}) \mathcal{B}_{kj} \rho^{(n-2)}_{k,j}(t) \right) \ & - i\alpha_1 \sum_k \left( \mathcal{A} \rho^{(n+1)}_k(t) + n_k \mathcal{C}_k \rho^{(n-1)}_k(t) \right) \end{aligned}$$ where the superoperators are:

• $$\mathcal{L}_\text{eff} \hat O = [ H_\text{eff}, \hat O ]$$, with $H_\text{eff} = H_S + \alpha_2 \hat x_B^2 \hat Q$,
• $\mathcal{A} \hat O = [\hat Q, \hat O]$,
• $$\mathcal{B}_{kj} \hat O = \eta_k \eta_j \hat Q \hat O - \eta_k^* \eta_j^* \hat O \hat Q$$,
• $$\mathcal{C}_k \hat O = \eta_k \hat Q \hat O - \eta_k^* \hat O \hat Q$$.

This equation recursively couples DDOs differing by $±1$ and $±2$ dissipatons, thus capturing the interplay of linear and quadratic system–bath coupling at all orders.

The non-Gaussian terms (those originating from quadratic coupling) cannot be captured by linear response; the DEOM explicitly encodes these effects through the GWT-2 algebraic structure.

5. Validation: Extended Zusman Equation and Consistency Tests

The validity of the dissipaton theory’s novel algebraic ingredients is established by an independent derivation of the “extended Zusman equation” in the high-temperature, overdamped (Smoluchowski) limit. The resulting equation for the conditional density operator $\hat \rho(x;t)$,

$$\partial_t \hat \rho(x; t) + (i\mathcal{L} + \hat L) \hat \rho(x;t) = - i [\alpha_1 x + \alpha_2 (x^2 - \eta_i^2 \partial_x^2)] [\hat Q, \hat \rho(x;t)] - 2 \eta_i \partial_x \{ (\alpha_1/2 + \alpha_2 x) \{ \hat Q, \hat \rho(x;t)\} \}$$

upon expansion in terms of Hermite polynomial eigenfunctions and mapping to a hierarchy over DDOs, is shown to exactly recover the DEOM structure when only a single exponential basis is present in the bath. This cross-verification demonstrates that the GWT-2 term introduced is both algebraically robust and physically correct, ensuring that the non-Gaussian dynamics is described self-consistently.

6. Extension to Higher-Order Nonlinear Bath Couplings and Practical Applications

The dissipaton algebra can be recursively extended to higher-order bath couplings (e.g., cubic, quartic), by formulating generalized Wick’s theorems GWT-$n$ for $n$-fold contractions. This sets the stage for systematically including arbitrary bath nonlinearities.

A principal practical application is to the calculation of optical absorption lineshapes for systems exhibiting quadratic bath coupling. In such models, the Hamiltonian is: $$H(t) = h_g |g\rangle\langle g| + ( h_e + \Delta h_B ) |e\rangle\langle e| - \hat\mu E(t)$$ with the bath difference given by: $$h_e - h_g = \alpha_0 + \alpha_1 \hat x_B + \alpha_2 \hat x_B^2$$ A polarization model determines $\alpha_1$ and $\alpha_2$ from solvation physics. Simulations show that pure linear coupling ($\alpha_2 = 0$) yields symmetric lineshapes, whereas nonzero quadratic coupling ($\alpha_2 \ne 0$), controlled via the ratio $\theta = \omega'/\omega$, induces marked asymmetry and skewness in the spectral profile—features uniquely attributable to non-Gaussian interference and inaccessible to linear response descriptions.

7. Impact and Significance

Dissipaton theory, as realized in the DEOM framework and its algebraic generalizations, delivers an exact and comprehensive solution for open quantum systems with nonlinear environmental couplings. The approach integrates rigorous algebraic development (generalized diffusion, non-Gaussian Wick’s theorems), independent physical validation (extended Zusman equation), and practical computational tractability (hierarchical DDOs, direct simulation of nonlinear bath effects). By facilitating the computation of real-time and spectral properties—including those governed by non-Gaussian bath physics—the dissipaton approach enables the quantitative and mechanistically resolved simulation of optical, transport, and relaxation processes in complex condensed-phase environments. The method’s ability to bypass the constraints of linear response and its extensibility to arbitrarily nonlinear baths mark it as a foundational framework for modern quantum dissipation studies in non-Gaussian environments (Xu et al., 2016).