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arXiv:2607.26093

Exact Finite-Dimensional Evaluation of Homodyne Quantum Trajectories for Single-Quadrature Hamiltonian

Jacob Emerson

quant-phSubmitted Jul 28, 2026• Announced Jul 29, 2026• 0 citations
v1 · Jul 28, 2026v2 · Aug 14, 2026
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Abstract

Simulation of nonlinear stochastic master equations generally requires trajectory-level evolution of large Hilbert-space density matrices, making strongly nonlinear continuously monitored systems computationally challenging. Here we identify an exactly solvable class of continuously monitored systems consisting of polynomial single-quadrature Hamiltonians, linear damping, and homodyne measurement of the same quadrature. For Gaussian initial states, we derive an exact finite-dimensional stochastic representation of the conditional dynamics for arbitrary finite-order quadrature moments $\langle Q^mP^n\rangle$ through closure of the conditional momentum hierarchy. The resulting coefficient equations depend only on the Hamiltonian degree and moment order, with computational complexity $\mathcal{O}(C_{d,n}T)$, independent of the Hilbert-space dimension. The method requires no Fock-space truncation, Gaussian approximation, or moment closure approximation. Numerical simulations demonstrate agreement with direct stochastic master equation evolution and establish an efficient exact method for studying nonlinear homodyne trajectories in this class of continuously monitored systems.

Comments
14 pages, 4 figures

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Simulation of nonlinear stochastic master equations generally requires trajectory-level evolution of large Hilbert-space density matrices, making strongly nonlinear continuously monitored systems computationally challenging. Here we identify an exactly solvable class of continuously monitored systems consisting of polynomial single-quadrature Hamiltonians, linear damping, and homodyne measurement of the same quadrature. For Gaussian initial states, we derive an exact finite-dimensional stochastic representation of the conditional dynamics for arbitrary finite-order quadrature moments ⟨QmPn⟩\langle Q^mP^n\rangle⟨QmPn⟩ through closure of the conditional momentum hierarchy. The resulting coefficient equations depend only on the Hamiltonian degree and moment order, with computational complexity O(Cd,nT)\mathcal{O}(C_{d,n}T)O(Cd,n​T), independent of the Hilbert-space dimension. The method requires no Fock-space truncation, Gaussian approximation, or moment closure approximation. Numerical simulations demonstrate agreement with direct stochastic master equation evolution and establish an efficient exact method for studying nonlinear homodyne trajectories in this class of continuously monitored systems.
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We derive the quantum filter for a quantum open system undergoing quadrature measurements (homodyning) where the input field is in a general quasi-free state. This extends previous work for thermal input noise and allows for squeezed inputs. We introduce a convenient class of Bogoliubov transformations which we refer to as balanced and formulate the quantum stochastic model with squeezed noise as an Araki-Woods type representation. We make an essential use of the Tomita-Takesaki theory to construct the commutant of the C*-algebra describing the inputs and obtain the filtering equations using the quantum reference probability technique. The derived quantum filter must be independent of the choice of representation and this is achieved by fixing an independent quadrature in the commutant algebra.
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