Tomogram-based quantifiers of nonclassicality dynamics in Kerr and cubic media

Tomogram-Based Quantifiers of Nonclassicality Dynamics in Kerr and Cubic Media

Introduction

The characterization and quantification of nonclassicality in continuous-variable quantum systems remain central challenges for both theoretical and experimental quantum optics. Nonclassical states—distinguished by phase-space features such as negativity in the Wigner function or sub-Poissonian statistics—are crucial resources for quantum information processing, metrology, and communication. Their reliable identification, particularly during their coherent evolution and under environmental decoherence, is essential for emerging quantum technologies. Many classical quantifiers, such as the Wigner negativity or Mandel’s $Q$-parameter, pose significant practical barriers due to experimental inaccessibility, non-uniqueness, or insensitivity to subtle quantum signatures, especially in the presence of realistic decoherence and in highly non-Gaussian regimes.

This paper introduces and quantitatively benchmarks tomogram-based measures—specifically, the homodyne nonclassical area and sum tomographic entropy—for tracking the real-time dynamics of nonclassicality in paradigmatic nonlinear photonic systems. These measures are directly accessible from balanced homodyne detection, obviating density matrix or full quasiprobability distribution reconstruction and thus offering a scalable route for experimental quantification. The analysis systematically addresses the evolution of coherent, photon-added coherent, and even coherent states in both Kerr and cubic nonlinear media, with explicit modeling of amplitude and phase damping via the Lindblad master equation.

Tomogram-Based Quantifiers: Formalism and Practicality

The central feature of the presented framework is the direct extraction of nonclassicality metrics from optical tomograms—positive marginalized distributions obtained from homodyne detection of quadrature phases. The homodyne nonclassical area is defined as the integrated excess quadrature variance relative to a coherent state, projected onto the optical tomographic plane. This is quantified as

$$\sigma(\ket{\psi}) - \sqrt{2}\pi = \int_{0}^{2\pi} \Delta X_{\theta} \, d\theta - \sqrt{2}\pi,\, \Delta X_{\theta} = \sqrt{\langle \mathbb{X}_\theta^2 \rangle - \langle \mathbb{X}_\theta \rangle^2}$$

where all moments are computed directly from the tomogram, bypassing density matrix inversion. For pure states, a nonzero value signals nonclassicality; for mixtures, this is only a sufficient indicator if classical mixing does not dominate.

Complementarily, the sum tomographic entropy across conjugate quadrature angles probes the information content and delocalization characteristics of the state, providing a sensitive diagnostic for higher-order fractional revivals and non-Gaussian phase-space interference, especially in dynamics where average quadrature variance-based indicators saturate.

Nonclassicality Dynamics in Kerr Media

Unitary Evolution and Fractional Revivals

For a Kerr Hamiltonian $H_{\mathrm{Kerr}} = \chi_1 a^{\dagger 2} a^2$ (with $[\hbar]=1$ and $\chi_1=5$), the non-equidistant spectrum supports quantum revivals and the formation of macroscopic superpositions (fractional revivals). For an initial coherent state, the homodyne nonclassical area is identically zero, but the evolution under Kerr nonlinearity generates pronounced, periodic excursions indicating the emergence of nonclassical features. For photon-added coherent and even coherent states, both inherently non-Gaussian, initial nonclassical areas are strictly positive. All classes show clear periodic recurrence of the nonclassical area at revival time $T_{\mathrm{rev}} = \pi/\chi_1$, serving as a precise marker for wave packet reconstruction. Dips in the nonclassical area correspond to instants of fractional revivals, revealing macroscopic superpositions.

Notably, the value and structure of both periodic revivals and fractional subpacket features are strongly field-strength dependent: increasing $|\alpha|^2$ systematically enhances quantum measures at each stage.

Decoherence: Amplitude and Phase Damping

Under amplitude damping (Kraus operator: $L = \sqrt{\gamma}a$), the nonclassical area decays monotonically and irreversibly toward zero, tracking population loss to the vacuum. In contrast, phase damping (Kraus operator: $L = \sqrt{\gamma} N$) preserves populations but suppresses coherences, resulting in decay of revival phenomena but with persistent local minima for weak coupling. This contrast is quantified: amplitude damping eliminates all signatures of revivals in the long-time limit, while phase damping preserves some vestige of near-revivals and superpositions over longer intervals.

Photon-added coherent states demonstrate enhanced robustness against decoherence, maintaining nonclassical area values higher than coherent or even coherent states at intermediate times.

Tomographic Entropy and Sensitivity to High-Order Fractional Revivals

Higher-order dynamical features invisible to variance-based metrics are exposed by the sum tomographic entropy $S(\theta) + S(\theta + \pi/2)$. In ideal unitary evolution, this entropy sum exhibits relative minima concomitant with each fractional revival and superposition event, regardless of initial state. Distinct minima mark the formation of multi-component Schrödinger cat-like states and are particularly pronounced for large photon numbers.

Environmental coupling attenuates these minima, with amplitude damping suppressing all revival signatures and driving entropy to vacuum-state values, while phase damping leaves some minima visible under weak coupling. The persistence and eventual disappearance of these entropy features function as quantitative measures of decoherence’s impact.

Nonclassicality Dynamics in Cubic Nonlinear Media

Cubic nonlinearity ($$\sigma(\ket{\psi}) - \sqrt{2}\pi = \int_{0}^{2\pi} \Delta X_{\theta} \, d\theta - \sqrt{2}\pi,\, \Delta X_{\theta} = \sqrt{\langle \mathbb{X}_\theta^2 \rangle - \langle \mathbb{X}_\theta \rangle^2}$$0) supports an even richer array of revival and superposition dynamics. For all considered initial states, the nonclassical area again reveals full and fractional revivals; notably, three-component fractional revivals at $$\sigma(\ket{\psi}) - \sqrt{2}\pi = \int_{0}^{2\pi} \Delta X_{\theta} \, d\theta - \sqrt{2}\pi,\, \Delta X_{\theta} = \sqrt{\langle \mathbb{X}_\theta^2 \rangle - \langle \mathbb{X}_\theta \rangle^2}$$1 and $$\sigma(\ket{\psi}) - \sqrt{2}\pi = \int_{0}^{2\pi} \Delta X_{\theta} \, d\theta - \sqrt{2}\pi,\, \Delta X_{\theta} = \sqrt{\langle \mathbb{X}_\theta^2 \rangle - \langle \mathbb{X}_\theta \rangle^2}$$2 are directly identified by recurrence of initial nonclassical area values.

Amplitude damping in the cubic medium enforces rapid decay to vacuum, and, as in the Kerr case, all quantum features are eradicated over long times. Phase damping, however, continues to suppress only the off-diagonal density matrix elements—the resulting steady state exhibits persistent quadrature variance contributions arising from statistical mixing, rendering the nonclassical area nonzero despite the absence of coherence.

Sum tomographic entropy metrics capture not only the main revivals but also intricate higher-order revival structures, again with entropy minima aligning precisely with macroscopic superposition formation and phase-space rotations. With moderate decoherence, only lower-order fractional revival features remain pronounced.

Implications and Future Directions

The results decisively demonstrate that tomogram-based measures address critical barriers in the operational quantification of nonclassicality dynamics: they are directly accessible via balanced homodyne detection—already standard in experimental quantum optics laboratories—eliminating the need for unstable or resource-intensive density matrix reconstruction. This renders them, in effect, scalable for real-time monitoring during unitary or open-system evolution, and flexible enough to handle a diverse range of initial states and highly non-Gaussian features.

The applicability of these measures extends beyond Kerr and cubic nonlinearities, suggesting deployment in the study of engineered dissipative processes, nonlinear oscillators in hybrid architectures, and the benchmarking of quantum error mitigation procedures.

On the theoretical axis, these results suggest refined approaches to quantum resource theory: since quantum features (e.g., multi-component cat states, squeezing, non-Gaussianity) can be tracked faithfully in experiment exclusively via tomograms, the operational meaning of such measures gains enhanced credibility.

The distinction observed between amplitude and phase damping channel responses provides actionable insight for dissipative engineering of quantum systems, including the design of nonclassical states resilient to specific types of environmental noise.

Future work should address the integration of tomogram-based quantifiers with machine learning approaches for automated state characterization, as well as the development of strict nonclassicality witnesses based on tomographic entropy dynamics. Additional exploration of multi-mode extensions and multimodal entanglement quantification—as well as the application to non-Gaussian state engineering—will further extend the practical impact.

Conclusion

This study establishes the homodyne nonclassical area and sum tomographic entropy as robust, experimentally viable quantifiers for tracking nonclassicality dynamics in nonlinear photonic media. These measures provide real-time access to the structure of quantum revivals, fractional superpositions, and the degradation of coherence under environmental interactions. This framework resolves several operational and scalability limitations of classical quantifiers and provides a foundation for next-generation quantum technologies requiring precise, inline diagnostics of quantum state dynamics.

Reference:

"Tomogram-based quantifiers of nonclassicality dynamics in Kerr and cubic media" (2605.03746)