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arXiv:2610.00667

Operator-language Feynman rules for driven-dissipative quantum systems: from mean field to non-Gaussian photon correlations

Peter Ehlers, Phi Hung Nguyen, Daniel Soh

quant-phSubmitted Sep 30, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

We construct Feynman rules for Lindblad master equations in the operator language of quantum optics: a fixed algebra of elementary operator moves and closed-form propagators for perturbation theory about an exactly solvable, in most cases Gaussian, generator. The expansion separates into a kinematic part (the shift algebra on the Wick-ordered eigenoperators $O_{mn}=:(c^\dagger)^m c^n:$, from which the vertex table for a specified interaction is assembled) and a dynamical part that enters only through the resolvent of the unperturbed generator, diagonal and additive for any stable, diagonalizable Gaussian $L_0$, including thermal and squeezed reservoirs and correlated decay, and equally available for a two-level emitter, driven or undriven. In the driven Kerr cavity the loop expansion is the semiclassical expansion and the series is asymptotic. In a strongly driven two-level atom, dressed lines replace a series of finite radius by one accurate deep into saturation, one line per Mollow component. For Kerr rings and chains the cost is polynomial in the number of sites at fixed perturbative order. On a three-site ring the diagrams recover the non-Gaussian part of the three-photon correlation that the Gaussian cumulant closure lacks; carried to tenth order they give the smallest $g^{(3)}$ error of the methods compared for $U\leq0.25κ$ at a cost below that of a fourth-order closure, and third- and fourth-order closures are more accurate at moderate coupling. The same rules, extended to the counting-field-tilted generator, give the photon-counting cumulants of a disordered eight-site chain beyond direct diagonalization; quantum-jump trajectories at three couplings agree with them to within $1.6$ standard errors, within one at the strongest coupling, where they place the third cumulant six standard errors above the Gaussian value.

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We construct Feynman rules for Lindblad master equations in the operator language of quantum optics: a fixed algebra of elementary operator moves and closed-form propagators for perturbation theory about an exactly solvable, in most cases Gaussian, generator. The expansion separates into a kinematic part (the shift algebra on the Wick-ordered eigenoperators Omn=:(c†)mcn:O_{mn}=:(c^\dagger)^m c^n:Omn​=:(c†)mcn:, from which the vertex table for a specified interaction is assembled) and a dynamical part that enters only through the resolvent of the unperturbed generator, diagonal and additive for any stable, diagonalizable Gaussian L0L_0L0​, including thermal and squeezed reservoirs and correlated decay, and equally available for a two-level emitter, driven or undriven. In the driven Kerr cavity the loop expansion is the semiclassical expansion and the series is asymptotic. In a strongly driven two-level atom, dressed lines replace a series of finite radius by one accurate deep into saturation, one line per Mollow component. For Kerr rings and chains the cost is polynomial in the number of sites at fixed perturbative order. On a three-site ring the diagrams recover the non-Gaussian part of the three-photon correlation that the Gaussian cumulant closure lacks; carried to tenth order they give the smallest g(3)g^{(3)}g(3) error of the methods compared for U≤0.25κU\leq0.25κU≤0.25κ at a cost below that of a fourth-order closure, and third- and fourth-order closures are more accurate at moderate coupling. The same rules, extended to the counting-field-tilted generator, give the photon-counting cumulants of a disordered eight-site chain beyond direct diagonalization; quantum-jump trajectories at three couplings agree with them to within 1.61.61.6 standard errors, within one at the strongest coupling, where they place the third cumulant six standard errors above the Gaussian value.
We develop a systematic projection-operator technique for constructing Gaussian approximations and their perturbative corrections in bosonic nonlinear models. As a case study, we apply it to the driven dissipative Kerr oscillator. In the absence of external driving, the model can be solved exactly within a low-dimensional Fock subspace, leading to strongly non-Gaussian states. Nevertheless, we demonstrate that the evolution of first- and second-order moments is captured by our Gaussian scheme with high accuracy even in this regime, providing a natural benchmark. For the general case with external driving, our approach reduces the equations of motion to a closed system for means and covariances and allows one to compute systematic corrections beyond the Gaussian level in closed form. We also calculate the dynamics of linear and quadratic combinations of creation and annihilation operators in both weak- and strong-drive regimes.
Motivated by correlated decay processes producing gain, loss, and lasing in driven semiconductor quantum dots [Phys. Rev. Lett. 113, 036801 (2014); Science 347, 285 (2015); Phys. Rev. Lett. 114, 196802 (2015)], we develop a theoretical technique by using Keldysh diagrammatic perturbation theory to derive a Lindblad master equation that goes beyond the usual second-order perturbation theory. We demonstrate the method on the driven dissipative Rabi model, including terms up to fourth order in the interaction between the qubit and both the resonator and environment. This results in a large class of Lindblad dissipators and associated rates which go beyond the terms that have previously been proposed to describe similar systems. All of the additional terms contribute to the system behavior at the same order of perturbation theory. We then apply these results to analyze the phonon-assisted steady-state gain of a microwave field driving a double quantum dot in a resonator. We show that resonator gain and loss are substantially affected by dephasing-assisted dissipative processes in the quantum-dot system. These additional processes, which go beyond recently proposed polaronic theories, are in good quantitative agreement with experimental observations.
We present a unified many-body perturbation theory for open quantum systems that treats dissipation, correlations, and external driving on equal footing. Using a Keldysh-Lindblad formalism, we introduce a diagrammatic treatment of dissipative interaction lines representing quasiparticle flows and fluctuations. Two new Feynman rules render the evaluation of dissipative diagrams compact and systematically improvable, while preserving the Keldysh and anti-Hermitian symmetries of the closed-system theory. Consequently, the structure of the Kadanoff-Baym equations remains unchanged, enabling existing numerical methods to be directly applied. To illustrate this, we derive dissipative versions of the second Born and GW approximations and identify the physical content of the self-energy components. Moreover, we demonstrate that time-linear approximations to the full Kadanoff-Baym equations retain their closed-form structure and can be efficiently used to simulate relaxation and decoherence dynamics. The impact of dissipation-induced correlations is illustrated in the driven Haldane model, where quasiparticles exhibit nontrivial stabilization and acquire lifetimes that far exceed those of the bare system. This framework establishes a general route toward first-principles modeling of correlated, driven, and dissipative quantum materials.
These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area which has applications across the quantum sciences dealing with systems subjected to external noise. The guiding idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes-Cummings model the reader will dis- cover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving (CPTP) maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.
We develop a recursive perturbative expansion for the time-convolutionless generator of an open quantum system in a generalized Lindblad form. This formulation provides a systematic approach to derive the generator at arbitrary order while preserving a Lindblad-like structure, without imposing assumptions on the system or environment beyond an initially uncorrelated state. The generator is written, at all orders, in a canonical form, which also corresponds to the minimal dissipation condition, which uniquely specifies the decomposition of the generator into Hamiltonian and dissipative contributions. To validate the method and show its effectiveness in addressing non-Markovian dynamics and strong-coupling effects, we compute the generator explicitly up to fourth order.
We combine the Briegel--Englert damping basis of a lossy cavity mode with the permutation-invariant Liouville basis of NNN identical, incoherently pumped two-level emitters. The resulting Liouvillian is exactly block tridiagonal in a single radial field index, with atomic blocks of dimension (N+33)\binom{N+3}{3}(3N+3​). The stationary state follows from a matrix continued fraction whose asymptotic closure is a linear matrix equation that we prove to be uniquely solvable throughout the dissipative regime A>0A>0A>0, and the same construction in the coherence-one sector yields the emission spectrum. The method is validated against direct diagonalization, weak-pump perturbation theory, and closed-system spectra, reaches N=16N=16N=16 in tens of seconds in the few-quanta regime, and has a quantified precision limit at large photon number. We use it to derive an exact two-atom photon-balance identity that isolates the cavity-induced pair coherence and relates it to a singlet--triplet population difference; to establish an exact pump-order hierarchy of connected correlations for a zero-temperature cavity and a parity property of the minimal collective-spin sector; and to benchmark the second-order cumulant closure used in few-emitter laser theory, which we find to be semi-quantitative for the photon number but to predict the wrong sign of the inter-emitter coherence above inversion. In a good-cavity regime the exact solution shows near-Poissonian statistics and strong line narrowing for two and three emitters.