Solving Caldeira-Leggett Model by Inchworm Method with Frozen Gaussian Approximation

Geshuo Wang; Siyao Yang; Zhenning Cai

doi:10.22331/q-2025-03-24-1667

Solving Caldeira-Leggett Model by Inchworm Method with Frozen Gaussian Approximation

Geshuo Wang¹, Siyao Yang², and Zhenning Cai³

¹Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
²Committee on Computational and Applied Mathematics, Department of Statistics, University of Chicago, Chicago, IL 60637 USA
³Department of Mathematics, National University of Singapore, Singapore 119076

Published:	2025-03-24, volume 9, page 1667
Editor:	Di Fang
Eprint:	arXiv:2408.01039v3
Doi:	https://doi.org/10.22331/q-2025-03-24-1667
Citation:	Quantum 9, 1667 (2025).

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Abstract

We propose an algorithm that combines the inchworm method and the frozen Gaussian approximation to simulate the Caldeira-Leggett model in which a quantum particle is coupled with thermal harmonic baths. In particular, we are interested in the real-time dynamics of the reduced density operator. In our algorithm, we use frozen Gaussian approximation to approximate the wave function as a wave packet in integral form. The desired reduced density operator is then written as a Dyson series, which is the series expression of path integrals in quantum mechanics of interacting systems. To compute the Dyson series, we further approximate each term in the series using Gaussian wave packets, and then employ the idea of the inchworm method to accelerate the convergence of the series. The inchworm method formulates the series as an integro-differential equation of “full propagators'', and rewrites the infinite series on the right-hand side using these full propagators, so that the number of terms in the sum can be significantly reduced, and faster convergence can be achieved. The performance of our algorithm is verified numerically by various experiments.

Featured image: Caldeira-Leggett model: a quantum particle coupled with many quantum harmonic oscillators.

Popular summary

Open quantum systems—where an interesting subsystem (such as spins, atoms, or electrons) interacts with an uninteresting environment (like electromagnetic fields or lattice vibrations)—are crucial in many fields, including quantum chemistry, quantum optics, and quantum computing. A fundamental example for studying such systems is the Caldeira-Leggett model, which describes a quantum particle coupled to a heat bath of quantum harmonic oscillators. Our goal is to study the reduced dynamics of the quantum particle while accounting for the dissipative effects of the environment.

As in many models of open quantum systems, a key challenge arises from the large number of degrees of freedom in the environment. Directly computing the full dynamics and then reducing to the interesting particle dimensions is computationally prohibitive. A common solution is to use perturbation theory, which expresses the reduced dynamics as an infinite summation of integrals represented by Feynman diagrams. By truncating this series and summing the diagrams, one can approximate the reduced dynamics without the curse of dimensionality. In this work, we build on this framework and develop a fast algorithm to efficiently compute the sum of relevant Feynman diagrams.

Our proposed fast algorithm relies on two key components. The first is the inchworm method, which resums Feynman diagrams to accelerate series convergence. This reformulation allows us to focus on lower-order renormalized Feynman diagrams, reducing both computational cost and numerical variance when evaluating the diagrams/integrals using Monte Carlo approximation. The second component is the frozen Gaussian approximation (FGA). In evaluating each renormalized diagram, the system-environment coupling is encoded in the particle positions, which are not directly accessible. The FGA, a Lagrangian-type method, approximates quantum dynamics by decomposing the system into Gaussian wavepackets with frozen widths. This enables us to efficiently approximate the particle positions using the centers of these wavepackets.

To our knowledge, this work represents one of the first attempts to numerically solve the real-time reduced dynamics of the Caldeira-Leggett model. Our framework opens the door to simulating more complex quantum phenomena, such as quantum tunneling effects, and provides a foundation for advancing numerical methods in open quantum system simulations.

► BibTeX data

@article{Wang2025solvingcaldeira,
  doi = {10.22331/q-2025-03-24-1667},
  url = {https://doi.org/10.22331/q-2025-03-24-1667},
  title = {Solving {C}aldeira-{L}eggett {M}odel by {I}nchworm {M}ethod with {F}rozen {G}aussian {A}pproximation},
  author = {Wang, Geshuo and Yang, Siyao and Cai, Zhenning},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1667},
  month = mar,
  year = {2025}
}

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[2] Olga Goulko, Hsing-Ta Chen, Moshe Goldstein, and Guy Cohen, "Transient dynamical phase diagram of the spin-boson model at finite temperature", Physical Review B 113 11, 115155 (2026).

[3] Yixiao Sun, Geshuo Wang, and Zhenning Cai, "Iterative Methods for Discrete Dyson Series and Applications in Open Quantum Systems", Journal of Chemical Theory and Computation (2026).

[4] Geshuo Wang, Yixiao Sun, Siyao Yang, and Zhenning Cai, "Accelerated inchworm method with tensor-train bath influence functional", Computer Physics Communications 325, 110164 (2026).

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