Dissipaton Equations of Motion (DEOM)

• DEOM is an exact nonperturbative framework for open quantum systems where a Gaussian environment is represented by dissipatons, offering both accurate reduced dynamics and physical bath interpretation.
• It employs a dissipaton algebra to construct a hierarchical structure equivalent to HEOM, while extending to phase-space, thermodynamic, and nonlinear coupling formulations.
• Applications span spectroscopy, transport, and impurity physics, enabling direct access to hybrid bath observables, system–bath correlations, and thermodynamic properties.

Searching arXiv for recent and foundational DEOM papers to ground the article. Dissipaton Equations of Motion (DEOM) is an exact, nonperturbative framework for open quantum systems in which a Gaussian environment is represented in terms of statistically independent quasi-particles called dissipatons. In this formulation, the bath is not treated merely as an eliminated influence or a hierarchy of auxiliary mathematical objects; rather, its fluctuations, hybrid bath observables, and system–bath entanglement are encoded in dissipaton density operators (DDOs), yielding a hierarchy that is formally equivalent to HEOM for reduced dynamics while providing a physically interpretable many-particle description of the environment (Wang et al., 2019). Subsequent developments established phase-space dissipaton algebra for bath coordinates and conjugate momenta, imaginary-time and nonequilibrium thermodynamic constructions, dissipaton-embedded master equations, and extensions to quadratic and other nonlinear environment couplings, thereby broadening DEOM from Gaussian linear-coupling dynamics to phase-space, thermodynamic, transport, spectroscopy, impurity, and selected nonlinear settings (Wang et al., 2022).

1. Origins and conceptual basis

DEOM was introduced to address the standard difficulty of open quantum dynamics: the full system-plus-bath Hamiltonian is exact but usually too large to solve directly, whereas approximate reduced-density-matrix methods often rely on weak coupling, Markovianity, or comparable simplifications (Wang et al., 2019). The framework retains the exactness of HEOM for Gaussian baths, but differs in physical interpretation: HEOM auxiliary operators are described as largely mathematical, while DEOM assigns explicit quasi-particle meaning to the bath degrees of freedom through dissipatons (Li et al., 2023).

The standard setting is a Hamiltonian of the form

$$H_{\rm tot}=H+h_{\rm bath}+H_{\rm sb},\qquad H_{\rm sb}=\hat Q \hat F,$$

or, equivalently in simpler notation,

$H = H_S + h_B + \hat Q \hat x,$

with linear system–bath coupling and a Gaussian or Wick-like environment characterized by its two-point correlation function and spectral density (Li et al., 2023, Wang et al., 2019). The equilibrium bath correlation is written as

$$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$

and DEOM begins by expressing this correlation as a finite sum of exponentials,

$$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$

with $\gamma_{\bar k}=\gamma_k^*$ when complex-conjugate pairs occur (Li et al., 2023).

Each exponential component is then assigned to a dissipaton, giving the decomposition

$\hat F=\sum_{k=1}^K \hat f_k,$

or, in solvation-coordinate notation,

$\hat x = \sum_{k=1}^K \hat f_k.$

The dissipatons satisfy pairwise correlations

$$\hat f_k(t)\hat f_{k'}(0)=\delta_{kk'}\eta_k e^{-\gamma_k t}, \qquad \hat f_{k'}(0)\hat f_k(t)=\delta_{kk'}\eta_{\bar k}^{*} e^{-\gamma_k t},$$

so each dissipaton behaves as an independent stochastic or Brownian-like mode with decay rate $\gamma_k$ and amplitude $\eta_k$ (Wang et al., 2019, Li et al., 2023). This quasi-particle picture is a defining conceptual feature of DEOM and underlies later interpretations of hybrid bath modes, bath moments, current operators, and thermodynamic observables (Wang et al., 2022).

2. Dissipaton algebra, DDOs, and the hierarchy

The fundamental dynamical variables of DEOM are the dissipaton density operators,

$H = H_S + h_B + \hat Q \hat x,$0

with $H = H_S + h_B + \hat Q \hat x,$1 and $H = H_S + h_B + \hat Q \hat x,$2; the zeroth-tier member $H = H_S + h_B + \hat Q \hat x,$3 is the reduced density operator (Wang et al., 2019, Li et al., 2023). In bosonic settings, $H = H_S + h_B + \hat Q \hat x,$4 denotes an irreducible or symmetrized product; in fermionic settings it becomes antisymmetric (Li et al., 2023, Su et al., 2023).

Closure of the hierarchy follows from the dissipaton algebra. One ingredient is the generalized diffusion relation,

$H = H_S + h_B + \hat Q \hat x,$5

which identifies dissipatons as generalized Brownian particles whose amplitudes relax diffusively (Li et al., 2023). The other ingredient is the generalized Wick theorem. For bosonic dissipatons, one representative relation is

$H = H_S + h_B + \hat Q \hat x,$6

with a corresponding right-action rule involving $H = H_S + h_B + \hat Q \hat x,$7 (Li et al., 2023). These operator identities permit exact tier-raising and tier-lowering actions of bath operators on DDOs.

The resulting DEOM hierarchy for Gaussian linear coupling is

$H = H_S + h_B + \hat Q \hat x,$8

or, in equivalent notations, the same tier structure with model-specific superoperators $H = H_S + h_B + \hat Q \hat x,$9 and $$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$0 (Wang et al., 2019, Li et al., 2023). For fermionic reservoirs the hierarchy acquires Grassmannian sign structure, for example

$$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$1

as used for electronic impurity problems (Cheng et al., 2018).

The formal equivalence between DEOM and HEOM for reduced dynamics is emphasized repeatedly, but DEOM differs by explicitly retaining bath quasi-particle content, thereby allowing direct access to hybrid system–bath observables, bath moments, and system–bath correlations (Li et al., 2023, Zhang et al., 2016). This also underlies the statement, made in multiple later works, that DEOM is not merely a solver for the reduced density matrix but a dissipaton-space mechanics of entangled system–environment dynamics (Wang et al., 2022).

3. Phase-space formulation and bath observables

A major extension of DEOM was the incorporation of the bath conjugate momentum, producing a phase-space dissipaton theory (Wang et al., 2019). Earlier DEOM formulations treated the bath coordinate $$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$2 and hybrid bath solvation coordinate dynamics, but not the conjugate momentum $$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$3. The phase-space construction introduces

$$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$4

where $$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$5 is the momentum counterpart of dissipaton $$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$6 (Wang et al., 2019).

The central momentum-action formula is

$$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$7

with the conjugate form

$$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$8

and $$\langle \hat F(t)\hat F(0)\rangle = \frac{1}{\pi}\int_{-\infty}^{\infty} d\omega\, \frac{e^{-i\omega t}J(\omega)}{1-e^{-\beta\omega}},$$9 (Wang et al., 2019). The construction was validated against two criteria. The necessary criterion is recovery of the canonical commutation relation,

$$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$0

and the sufficient criterion is the operator-action identity

$$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$1

for arbitrary $$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$2 (Wang et al., 2019).

The explicit commutator algebra confirms the canonical structure. For example,

$$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$3

and summing over indices with

$$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$4

recovers

$$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$5

exactly in DEOM space (Wang et al., 2019).

This phase-space extension is practically important because current and transport observables often depend directly on bath momentum. In the heat-current example,

$$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$6

so evaluation of the equilibrium current autocorrelation

$$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$7

requires the momentum algebra (Wang et al., 2019). More broadly, the Perspective on dissipaton theories explicitly describes DEOM as a phase-space, quasi-particle reformulation that supports Schrödinger-picture, Heisenberg-picture, and imaginary-time constructions in a unified setting (Wang et al., 2022).

A related development is the dissipaton-embedded quantum master equation (DQME), which maps each dissipaton to a continuous real variable $$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$8 through

$$\langle \hat F(t)\hat F(0)\rangle=\sum_{k=1}^K \eta_k e^{-\gamma_k t}, \qquad \langle \hat F(0)\hat F(t)\rangle=\sum_{k=1}^K \eta_{\bar k}^{*} e^{-\gamma_k t},$$9

and encodes the hierarchy in a single distribution

$\gamma_{\bar k}=\gamma_k^*$0

The resulting DQME,

$\gamma_{\bar k}=\gamma_k^*$1

with

$\gamma_{\bar k}=\gamma_k^*$2

gives a Smoluchowski-type interpretation of dissipatons as generalized Brownian particles (Li et al., 2023). This suggests that DEOM can be regarded either as a hierarchy of DDOs or as a continuous dissipaton-coordinate dynamics, with exact equivalence at the level of the chosen bath decomposition.

4. Imaginary-time DEOM and thermodynamic formulations

DEOM was subsequently generalized from real-time dynamics to equilibrium and transient thermodynamics. One development introduced an imaginary-time DEOM via analytical continuation $\gamma_{\bar k}=\gamma_k^*$3, yielding equilibrium DDOs and the hybridization partition function (Gong et al., 2020). In this construction, the imaginary-time density operator

$\gamma_{\bar k}=\gamma_k^*$4

is projected into dissipaton space,

$\gamma_{\bar k}=\gamma_k^*$5

and propagation to $\gamma_{\bar k}=\gamma_k^*$6 yields equilibrium DDOs and the hybridization partition function $\gamma_{\bar k}=\gamma_k^*$7 (Gong et al., 2020).

This enables direct evaluation of the hybridization Helmholtz free energy,

$\gamma_{\bar k}=\gamma_k^*$8

for isothermal mixing of a local impurity and a Gaussian bath (Gong et al., 2020). A thermodynamic integration formalism is also used, with $\gamma_{\bar k}=\gamma_k^*$9-dependent coupling turned on from $\hat F=\sum_{k=1}^K \hat f_k,$0 to $\hat F=\sum_{k=1}^K \hat f_k,$1, so that equilibrium DDOs determine the reversible work and hence the free-energy change (Gong et al., 2020, Wang et al., 2022).

The same framework was extended to nonequilibrium thermodynamics through $\hat F=\sum_{k=1}^K \hat f_k,$2-DEOM, permitting work-distribution calculations and numerical verification of the Jarzynski equality and Crooks relation (Wang et al., 2022). The Perspective states that dissipaton theories were developed for studying equilibrium and nonequilibrium thermodynamic mixing processes and that the Jarzynski equality and Crooks relation are accurately reproduced numerically (Wang et al., 2022).

Within the spin–boson illustration of the unified thermodynamic DEOM, the hybridization free energy is negative and decreases monotonically, consistent with spontaneity, while hybridization internal energy and entropy show turnover behavior as functions of temperature (Gong et al., 2020). The same work distinguishes the thermodynamic entropy from the reduced-state von Neumann entropy and reports a transient negative hybridization entropy interpreted as an indication of “solvent-cage” formation (Gong et al., 2020). These statements concern the specific thermodynamic analysis of that paper rather than a universal property of all DEOM applications.

A plausible implication is that dissipaton-space thermodynamics is not an auxiliary add-on but a direct consequence of retaining explicit system–bath hybrid variables. That implication is consistent with later presentations that place real-time DEOM, imaginary-time DEOM, and $\hat F=\sum_{k=1}^K \hat f_k,$3-based thermodynamic constructions within one formal family (Wang et al., 2022).

5. Extensions beyond Gaussian linear coupling

A central line of development concerns nonlinear environmental backactions, especially quadratic bath couplings. A 2016 generalization formulated DEOM for a non-Gaussian environment with Hamiltonian

$\hat F=\sum_{k=1}^K \hat f_k,$4

arguing that, unlike the linear bath-coupling case, the influence of a non-Gaussian environment cannot be completely characterized with the linear response theory (Xu et al., 2016). The key algebraic advance is a generalized Wick theorem with dissipaton-pair contributions, needed because $\hat F=\sum_{k=1}^K \hat f_k,$5 introduces operator products beyond single contractions (Xu et al., 2016).

The resulting extended hierarchy contains not only the usual $\hat F=\sum_{k=1}^K \hat f_k,$6 couplings from linear terms but also $\hat F=\sum_{k=1}^K \hat f_k,$7 channels and same-tier quadratic contributions. A representative form is

$\hat F=\sum_{k=1}^K \hat f_k,$8

with superoperators defined by $\hat F=\sum_{k=1}^K \hat f_k,$9, $\hat x = \sum_{k=1}^K \hat f_k.$0, and $\hat x = \sum_{k=1}^K \hat f_k.$1 (Xu et al., 2016). A later comprehensive account emphasized that the extended DEOM is verified by an independent Brownian-solvation-mode embedded hierarchical quantum master equation and by thermodynamic checks reproducing Jarzynski and Crooks (Chen et al., 2022).

An alternative exact route to nonlinear coupling is the stochastic-fields-dressed DEOM (SFD-DEOM), in which the nonlinear bath term is decoupled by a Hubbard–Stratonovich transformation so that only linear bath coupling remains for each stochastic realization (Chen et al., 2021). For the quadratic-coupling Hamiltonian

$\hat x = \sum_{k=1}^K \hat f_k.$2

the transformed stochastic Hamiltonian becomes

$\hat x = \sum_{k=1}^K \hat f_k.$3

with an analogous bra-side expression involving $\hat x = \sum_{k=1}^K \hat f_k.$4 (Chen et al., 2021). The hierarchy is then of ordinary DEOM form but with stochastic system operators, and the exact reduced density is obtained after ensemble averaging (Chen et al., 2021).

More recent extended dissipaton theories treat quadratic fermionic couplings in electronic impurity systems. One such hierarchy for quadratic fermionic bath operators contains both explicit $\hat x = \sum_{k=1}^K \hat f_k.$5 and $\hat x = \sum_{k=1}^K \hat f_k.$6 couplings and was applied to the Kondo impurity model (Su et al., 2023). Another presentation unified linear and quadratic fermionic couplings in an extended DEOM and equivalent dissipaton-embedded master equation, with application to an adatom–graphene composite (Su et al., 2024). In both cases, the extension is described as exact, non-Markovian, and non-perturbative within the bath-correlation decomposition.

These developments clarify a recurring misconception. Standard DEOM is exact for Gaussian baths with linear coupling after exponential decomposition, but nonlinear or quadratic environmental couplings require additional algebraic structure, stochastic dressing, or an extended hierarchy. The later theories are extensions of DEOM, not automatic consequences of the original linear-coupling formalism (Xu et al., 2016, Chen et al., 2021, Chen et al., 2022).

6. Applications across spectroscopy, transport, impurity physics, and anharmonic environments

DEOM has been applied across a wide range of open-system problems in which exact non-Markovian dynamics and explicit bath observables are advantageous.

In excitonic and vibronic spectroscopy, DEOM was used to study Herzberg–Teller vibronic coupling in excitation energy transfer and two-dimensional coherent spectroscopy (Zhang et al., 2016). There the transition dipole depends linearly on the bath coordinate,

$\hat x = \sum_{k=1}^K \hat f_k.$7

so optical observables explicitly involve hybrid bath dynamics (Zhang et al., 2016). The simulations show that non-Condon vibronic coupling intensifies dynamical electronic-vibrational energy transfer, enhances total system-and-bath quantum coherence, and enriches the interpretation of nonlinear spectroscopy (Zhang et al., 2016).

In photosynthetic reaction-center modeling, a mixed DEOM–Lindblad strategy treated the phonon bath exactly with DEOM and the photon bath by a Lindblad dissipator (Fang et al., 2022). The mixed scheme showed transfer coherence up to a few hundreds femtoseconds and an environmental manipulation effect on current that differs qualitatively from fully Markovian or classical-bath descriptions (Fang et al., 2022). This application uses DEOM primarily to preserve non-Markovian phonon memory and complex bath correlations.

In quantum impurity and Kondo physics, DEOM has been used as a numerically exact open-system method for triple quantum dots, double quantum dots, and impurity models (Cheng et al., 2018, Chen et al., 2023). In the open triple quantum dot problem, DEOM was used to analyze long-range entanglement of Kondo clouds through spectral functions, spin–spin correlations, and occupancies, with a conduction-electron-like peak on the intermediate dot interpreted as a signature of overlap between two Kondo clouds (Cheng et al., 2018). In the double-impurity Anderson model, DEOM was employed to compute both impurity spectral functions and current-noise spectra in the Kondo regime, with the Kondo signatures reshaped by inter-dot Coulomb interaction, inter-dot hopping, and chemical-potential bias (Chen et al., 2023). In the quadratic-coupling Kondo impurity model treated by extended DEOM, a sharp Kondo peak develops near the Fermi energy at low temperature, while perturbation theory fails and can violate the Friedel sum rule (Su et al., 2023).

In transport and current-correlation problems, DEOM has also been adapted to many-level electron–phonon dynamics. For a one-dimensional Peierls-type model with Brownian-oscillator spectral density, DEOM was used to compute the finite-temperature real-time current autocorrelation function, optical conductivity, diffusion constant, and mobility (Janković, 20 Aug 2025). That work exploited dissipaton algebra to handle the phonon-assisted current operator and used a momentum-space reformulation to reduce computational cost (Janković, 20 Aug 2025). The reported conclusion is that with increasing damping, DEOM optical-conductivity profiles become qualitatively similar to transient-localization predictions, suggesting that low-frequency enhancements found for undamped vibrations are artifacts of the undamped limit (Janković, 20 Aug 2025).

In spin-lattice and spin-environment problems, DEOM has been generalized in two distinct directions. One work treated a continuous bath of independent spins by mapping the spin bath, in its linear-response limit, onto an effective Gaussian environment characterized by a temperature-dependent spectral density (Ying et al., 2023). Another used extended DEOM for spin-lattice relaxation with linear and quadratic phonon couplings and showed that exact non-Markovian rate kernels can deviate substantially from Fermi’s golden rule, especially for quadratic coupling where the kernel exhibits a free-induction-decay-like feature and coupling-dependent damping (Bi et al., 2024).

In molecular spectroscopy with structured correlated environments, ext-DEOM was used to treat correlated vibration–solvent effects and Duschinsky rotations in electron transfer and optical spectra (Zhu et al., 10 Mar 2025). There the system–bath coupling takes the quadratic form

$\hat x = \sum_{k=1}^K \hat f_k.$8

and the hierarchy contains $\hat x = \sum_{k=1}^K \hat f_k.$9 channels from linear terms and $$\hat f_k(t)\hat f_{k'}(0)=\delta_{kk'}\eta_k e^{-\gamma_k t}, \qquad \hat f_{k'}(0)\hat f_k(t)=\delta_{kk'}\eta_{\bar k}^{*} e^{-\gamma_k t},$$0 channels from quadratic terms (Zhu et al., 10 Mar 2025). The spectra show that Duschinsky rotation and solvent-induced correlations can substantially reshape peak positions, widths, and intensities (Zhu et al., 10 Mar 2025).

Finally, a dissipaton-based approach to a locally probed one-dimensional chain formulated the dynamics directly in terms of c-number dissipaton moments rather than density operators, with cross-tier recursive couplings for higher-order chain–probe interactions (Qi et al., 31 Mar 2026). This suggests a further methodological branch in which the dissipaton algebra is retained but the propagated objects are ordinary moments tailored to transport observables such as heat current.

Taken together, these applications indicate that DEOM functions as a general exact hierarchy formalism for Gaussian baths and as a platform for several controlled extensions. Its distinctive strength lies not only in exact reduced dynamics but in direct access to hybrid bath coordinates, momenta, current operators, thermodynamic quantities, and correlated system–bath observables across spectroscopy, transport, impurity physics, and selected nonlinear environments (Wang et al., 2022).