Tensor-network approach to quantum optical state evolution beyond the Fock basis

TL;DR

The paper presents a continuous-variable, tensor-network approach using matrix product states (MPS) to simulate nonlinear optical quantum state evolution, addressing the exponential resource growth tied to photon number. By formulating SPDC in the quadrature basis and solving the discretized Schrödinger equation with a DMRG-like optimizer, the method achieves accurate dynamics while maintaining compact representations through MPS/MPO, enabling high-intensity pump scenarios ($α$ up to $100$) that are intractable with Fock-basis methods. Validation includes energy conservation, pump depletion benchmarks, and quadrature squeezing analyses, with strong fidelity to Fock-basis results at moderate amplitudes and robust sanity checks at large amplitudes. This framework offers a scalable route to modeling multimode quantum light and nonlinear optical phenomena beyond traditional approaches, with potential extensions to multimode networks and higher-order nonlinearities using time-dependent variational principles in MPS and alternative tensor-network architectures.

Abstract

Understanding the quantum evolution of light in nonlinear media is central to the development of next-generation quantum technologies. Yet modeling these processes remains computationally demanding, as the required resources grow rapidly with photon number and phase-space resolution. Here we introduce a tensor-network approach that efficiently captures the dynamics of nonlinear optical systems in a continuous-variable representation. Using the matrix product state (MPS) formalism, both quantum states and operators are encoded in a highly compressed form, enabling direct numerical integration of the Schrödinger equation. We demonstrate the method by simulating degenerate spontaneous parametric down-conversion (SPDC) and show that it accurately reproduces established theoretical benchmarks - energy conservation, pump depletion, and quadrature squeezing - even in regimes where conventional Fock-basis simulations become infeasible. For high-intensity pump fields ($α= 100$), the MPS representation achieves compression ratios above $3\cdot 10^3$ while preserving physical fidelity. This framework opens a scalable route to modeling multimode quantum light and nonlinear optical phenomena beyond the reach of traditional methods.

Figures (4)

• Figure 1: Encoding the wave function of the entire system in the MPS representation. Wave function of each mode is discretized with the required precision. Then it is converted to the MPS format. After that, the MPS of the entire system is assembled from stacking MPS tensors of each mode. In this example, the wave function of the pump mode is determined on the segment $R^{(p)} = [-8,8]$, signal mode - on the segment $R^{(s)} = [-9,4]$. Each mode is discretized with N = 16 points which corresponds to a MPS with 4 tensors (for each mode). The MPS of entire system contains 8 tensors.
• Figure 2: (a,b) Photon population dynamics $\langle n\rangle=\text{Tr}\left(\rho a^\dagger a\right)$ in the pump and signal modes (left Y-axis) and bond dimension change (right Y-axis) during quantum evolution in the SPDC process. Blue and red solid lines correspond to the MPS algorithm; green line shows total energy of the system expressed in the number of signal photons; orange solid line demonstrates how the bond dimension changes during the evolution. (a) Pump mode is initialized in the coherent state $\alpha = 10$. Crossed dots demonstrate state vector simulation in Fock basis. (b) Pump mode is initialized in the coherent state $\alpha = 100$. Black dashed line shows the theoretical limit of pump mode depletion. (c) Fidelity between Fock-basis state-vector and continuous-basis MPS simulations of the signal and pump density matrices during the quantum evolution. Fidelity calculation is done in the Fock basis. (d) Variance of $X$-quadrature in the signal mode during quantum evolution with the pump mode initialized at $\alpha=10$ and $\alpha=100$. The blue and red lines demonstrate MPS approach, while orange and green lines correspond to the theoretical analysis given in Ref. kinsler1993limits. The black crosses indicate variance for the ideal single-mode squeezing in the parametric approximation (pump mode is treated as a classical field).
• Figure 3: (a)$l_2$ residual norm $||\tilde{\psi_t}-U\tilde{\psi}_{t+1}||/\sqrt{\mathcal{N}}$ of the linear system (\ref{['eq:diffequation']}) during the quantum evolution simulation, for pump mode initialized at $\alpha=10$ ($\mathcal{N}=1$) and $\alpha=100$ ($\mathcal{N}=2^5$). (b) Inverse compression ratio. The ratio between the total number of elements in MPS and the mesh size throughout the SPDC evolution. Blue (red) line corresponds to the case when the pump mode is initialized in a coherent state $\alpha = 10$ ($\alpha = 100$).
• Figure 4: Diagrammatic illustration of DMRG style algorithm for solving linear systems in the MPS format. (a) System of linear equations in the MPS form; (b) Functional which global minimum corresponds to the solution of a linear system of equations. (c) Variation of the functional with respect to a pair of MPS tensors. (d) Local linear system for the corresponding pair of tensors.