Wigner's Phase Space Current for Variable Beam Splitters -- Phase Space Rotations and Newtonian Trajectories

Abstract: Beam splitters allow us to superpose two continuous single mode quantum systems. To study the behaviour of beam splitters' strongly mode mixing dynamics we consider variable beam splitters acting on Wigner's phase space distribution, W , the evolution of which is governed by the continuity-equation {\partial τ} W = - {\nabla} J. We derive the form of the corresponding Wigner current, J. J's form allows us to use a classical trajectories-approach to analyze the influence of the two modes on each other. We show that the dynamics for variable beam splitters amounts to a rotation confined within the plane of the two positions together with the same simultaneous rotation confined within the plane of the two momenta. In this way explicit and very transparent expressions for the rotated Wigner distributions and Wigner currents can be given in terms of classical trajectories. This helps us to gain deeper insights and perform geometrical analyses of the mixing of modes at beam splitters.

• The paper demonstrates that two-mode beam splitter dynamics can be mapped exactly to classical, rotation-like trajectories in phase space.
• It utilizes the Moyal bracket and symplectic transformations to show conservation of phase space volume and quantum-classical correspondence.
• The framework offers practical tools for quantum state tomography and entanglement measurement by reconstructing input states from measured quadratures.

Wigner's Phase Space Current for Variable Beam Splitters: Formal Analysis and Trajectory-Based Insights

Introduction

This paper develops a comprehensive phase-space-based framework for describing quantum optical beam splitters using Wigner's phase space distribution and, crucially, the associated Wigner phase space current. Contrary to the conventional focus on quantum states in Fock or abstract Hilbert space, the analysis leverages the Wigner current formalism to elucidate the mode-mixing dynamics of variable beam splitters. Special emphasis is placed on the explicit geometric and dynamical structure—demonstrating that, for perfect lossless beam splitters, the evolution throughout phase space is entirely analogous to well-defined, rotation-like Newtonian trajectories.

Theoretical Framework

The authors' formalism is grounded in the Wigner–Moyal phase space representation for continuous-variable two-mode quantum systems. The state is encoded in a four-dimensional Wigner distribution $W(\vec{r},T)$ with coordinates $\vec{r} = (x_a,p_a,x_b,p_b)$. The evolution is cast as a continuity equation involving the phase space current $\vec{J}$, governed by an effective time-parametrized beam splitter Hamiltonian $H_M$ that is bilinear in bosonic mode operators.

The explicit use of the Moyal bracket formulation allows rigorous mapping between the Hilbert space commutator and phase-space flows. The dynamical equations are shown to reduce to classical (i.e., Hamiltonian) form due to the quadratic structure of $H_M$—hence, the quantum–classical correspondence becomes exact for the two-mode beam splitter setting.

Newtonian Trajectories and Phase Space Rotations

The authors solve the Wigner current continuity equation analytically, yielding a set of rotation-like transformations in the joint $(x_a, x_b)$ and $(p_a, p_b)$ position and momentum planes. The mode mixing implemented by the beam splitter corresponds mathematically to classical phase space rotations with fictitious time parameter $\mathcal{T}$ determined by device transmissivity.

Specifically, the output Wigner distribution is expressed as a simple reparametrization of the input under these rotations:

$$W(\vec{r}',\mathcal{T}) = W(\mathcal{R}(\vec{r},\mathcal{T}), 0)$$

where $\mathcal{R}$ is the explicit symplectic rotation mapping input to output quadratures. These results hold for arbitrary (not necessarily separable) input states and for arbitrary variable beam splitter angles.

The framework immediately implies conservation of phase space volume (Liouvillian flow), since the transformations are symplectic with vanishing divergence in the velocity field. Remarkably, this means that all figures of merit dependent on the full joint Wigner function—such as total entanglement and overall nonclassicality—are invariant under the beam splitter's action.

Implications for Quantum Measurement and Tomography

Connections are established between the phase space framework and experimental quantum state tomography using balanced homodyne detection (BHD). The mapping of measured quadrature distributions before and after the beam splitter is explicitly described by the symplectic relations, emphasizing that post-processing can be used to reconstruct pre-mixing correlations provided the full measurement 4-tuple is retained.

Additionally, for initial product states, the evolution after the beam splitter—which results in entangled outputs—remains tractable within this geometric picture. In cases where experimentalists seek to interpret or invert beam splitter action, the formalism provides operational recipes for data remapping, thereby facilitating analysis of post-selected or conditional experiments.

Single-Mode Marginals and Nonclassicality

A critical finding is the contrast between the tractable, classical-like behavior of the full two-mode system and the much more complex dynamics exhibited by single-mode marginals after tracing out the other mode. Single-mode reduced Wigner functions can display highly nonclassical behaviors, including singular changes in phase space volume, in stark contrast to the regular, Liouvillian character of the complete two-mode dynamics. This distinction delineates the operational boundary where quantum effects such as decoherence, squeezing, and mode entanglement introduce substantial deviations from classical mechanics.

Broader Context and Future Directions

The results provide a transparent geometric perspective on quantum optical networks containing beam splitters, enabling direct visualization and calculation of mode mixing effects at the level of phase space currents. The approach is particularly valuable in the design and interpretation of experiments where continuous variable entanglement, squeezing, and non-Gaussianity are central.

From a theoretical standpoint, the fact that the classical description is exact only for the full two-mode system but breaks down in reduced single-mode representations points toward further investigations into the emergence of nonclassical phenomena upon subsystem tracing and decoherence. Extensions to open quantum systems, non-Gaussian resources, and networks with more general non-bilinear interactions are natural avenues for future research.

Conclusion

This work establishes a rigorous, analytic, and geometrically insightful description of variable beam splitter dynamics in quantum optics using the Wigner phase space current formalism. The formalism highlights the symmetry and simplicity of the two-mode phase space evolution—showing that the entire process is equivalent to classical symplectic rotations along Newtonian trajectories, with strict phase space volume conservation. The analysis draws a sharp distinction between global, regular evolution and local, potentially singular behavior of single-mode marginals. These results have direct implications for quantum state reconstruction, entanglement measurement, and the foundational understanding of quantum–classical correspondence in CV systems.