We study the dynamics of light interacting with a near-resonant atomic medium using the truncated Wigner and positive P phase-space representations. The atomic degrees of freedom are described using the Jordan-Schwinger mapping. The dynamics is first analyzed under unitary evolution and subsequently in the presence of an optical reservoir. While both approaches capture the main features of the light-matter dynamics, we find that the truncated Wigner approximation exhibits noticeable deviations for stronger interaction strengths and when reservoir-induced noise becomes significant.
The phase-space formulation of quantum mechanics [1][2][3][4][5][6][7][8][9][10] provides a conceptually transparent and self-contained alternative to the conventional Hilbert-space description. In this approach, quantum states are represented by quasiprobability distributions defined over a classical phase space, observables are mapped to 𝑐-number functions, and quantum dynamics is reformulated in terms of stochastic evolution . This framework offers not only an intuitive visualization of quantum states and processes, but also practical computational tools for regimes in which the direct solution of Heisenberg equations of motion or master equations becomes prohibitively complicated [11][12][13].
For bosonic systems, several phase-space representations are widely employed, most notably the Wigner (𝑊) [14], Glauber-Sudarshan (𝑃) [15,16], and Husimi (𝑄) [17,18] distributions. Despite their broad applicability, each of these representations suffers from significant limitations [19]. In particular, some distributions become highly singular for physically relevant quantum states; for instance, the Glauber-Sudarshan 𝑃 function is more singular than a delta function for squeezed states and number eigenstates. Other representations, such as the Wigner function, may take negative values and therefore cannot be interpreted as genuine probability densities, but rather as quasiprobability distributions. Furthermore, the equations governing their time evolution are not always true Fokker-Planck equations [20]; they may involve derivatives of order higher than two or, even when second order, fail to possess a positive-definite diffusion matrix.
To overcome these difficulties and enhance numerical applicability, particularly for complex and dissipative systems, several modified phase-space techniques have been developed [21,22]. Among the most widely used are the positive 𝑃 representation (PPR) and the truncated Wigner approximation (TWA), each addressing different aspects of the challenges outlined above.
The PPR, introduced by Drummond and Gardiner [23], overcomes several limitations of earlier phase-space methods. When the resulting evolution equation is of second order, the diffusion matrix can always be chosen positive definite, yielding a bona fide Fokker-Planck equation. Consequently, the probability distribution remains positive at all times, provided it is initialized with a suitable positive distribution, a condition that can always be satisfied [24,25]. These benefits come at the cost of doubling the number of phase-space variables, thereby increasing the dimensionality of the problem. More critically, the enlarged phase space gives rise to numerical instabilities driven by the interplay between stochastic noise and nonlinear dynamics. Random fluctuations can displace trajectories away from the stable “classical” manifold into unstable regions, where nonlinear drift terms strongly amplify deviations. This mechanism may lead to large excursions and, in extreme cases, to movable singularities in which trajectories diverge in finite time. Additional nonlinearities further exacerbate this behavior by opening new channels for amplification. Despite these challenges, the PPR has been widely and successfully applied in quantum optics [26][27][28] and ultracold atomic physics [29][30][31].
The TWA [32,33] provides a complementary approach that is particularly well suited to regimes close to the classical limit, where quantum fluctuations are small. To leading order, quantum effects enter only through the Wigner distribution of the initial conditions, while the subsequent dynamics are governed by classical equations of motion. This approximation has therefore been employed to study semiclassical phenomena in ultracold atomic systems [34][35][36][37]. Its practical efficiency, however, relies on a systematic truncation, which can lead to quantitative inaccuracies in regimes of strong interactions, long evolution times, or small particle numbers.
Broadly speaking, the regimes of validity of the PPR and the TWA are largely complementary: the TWA is typically accurate in the presence of strong driving, whereas the PPR remains stable under strong dissipation [38]. This complementarity has motivated numerous comparative studies of the two techniques [39][40][41][42][43]. Nevertheless, a comprehensive and systematic benchmark of these methods in open, strongly interacting light-matter systems (where intrinsic quantum fluctuations and reservoir-induced noise are essential) has not yet been performed. Addressing this gap constitutes the central objective of this paper.
To this end, we consider a quantum model describing the arXiv:2602.17660v1 [quant-ph] 19 Feb 2026 propagation of radiation in an atomic two-level medium subject to radiative damping [44]. Using the PPR, we have previously applied this framework to the study of self-induced transparency (SIT) [45] in gas-filled hollow-core photonic crystal fibers, demonstrat
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