Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2503.20699 · Journal of High Energy Physics

Quantum decoherence in the Caldeira-Leggett model by the real-time path integral on a computer

Jun Nishimura, Hiromasa Watanabe

Published Sep 24, 2025• 2 citations
arXivHTMLPDF

Abstract

A bstract We propose first-principle calculations of an open system based on the real-time path integral formalism treating the environment as well as the system of our interest together on a computer. The sign problem that occurs in applying Monte Carlo methods can be overcome in general by using the so-called Lefschetz thimble method, which has been developed over the past decade. Here we focus on the Caldeira-Leggett model, which is well known, in particular, as a model of quantum decoherence. In this case, the calculation simplifies drastically since the path integral becomes Gaussian for typical initial conditions. The relevant saddle point, which is unique and complex, can be determined by solving a linear equation with a huge but sparse coefficient matrix, and the integration over the Lefschetz thimble can be performed analytically. Thus we obtain, without assumptions or approximations, the reduced density matrix after a long-time evolution, tracing out a large number of harmonic oscillators in the environment. In particular, we confirm the dependence of the decoherence time on the coupling constant and the temperature that has been predicted from the master equation in a certain parameter regime.

DOI
10.1007/jhep09(2025)197

Related papers

Numerical studies on real time evolution in quantum systems - Quantum decoherence and the beginning of the universe

Naoyuki Yamamori

Real-time path integrals provide a natural formulation of quantum dynamics also in situations where a Hamiltonian description is unavailable or impractical, but their direct evaluation is obstructed by severe phase cancellation from the oscillatory weight.A guiding theme of this thesis is that, after complexifying the integration variables, real-time path integrals are naturally organized by complex saddle points.In favorable cases the original contour can be replaced by steepest-descent manifolds (Lefschetz thimbles) attached to the relevant saddles, providing a controlled saddle-based description of real-time observables.In Part I, we study quantum decoherence in a closed-system real-time path-integral formulation: the total system (subsystem plus environment) evolves unitarily, and the environment is traced out only at the final time to obtain the reduced density matrix.For the Caldeira-Leggett model with Gaussian initial conditions, the discretized effective action becomes a quadratic form, so that the reduced density matrix is characterized by a complex saddle point and the thimble integral reduces to a finite-dimensional Gaussian integral.The complex saddle is obtained by solving a large but sparse linear system, allowing an exact evaluation of the reduced density matrix without relying on the usual master-equation approximations.We then extend this methodology to a three-dimensional cubic lattice toy model by separating center-of-mass and relative coordinates, treating the center-of-mass mode as the subsystem and the relative modes as an effective environment.This enables controlled numerical studies of how the suppression of off-diagonal matrix elements depends on the coupling, inverse temperature, and system size, and allows us to extract the early-time scaling behavior of decoherence diagnostics from the complex-saddle evaluation.In Part II, we investigate spacetime dynamics in the Lorentzian IKKT matrix model.To define a Lorentzian formulation that is not continuously connected to the Euclidean model by the standard contour deformation, we include a Lorentz invariant mass term with coefficient γ, for which the classical equations admit real, smooth, expanding solutions when γ > 0. Since the Lorentz group is noncompact, the path integral can contain a divergent volume from the boost directions; we remove this volume nonperturbatively by implementing a boost "gauge-fixing" that selects the frame minimizing tr(A 0 ) 2 , leading to the condition tr(A 0 A i ) = 0 with the associated Faddeev-Popov determinant.We then apply the complex Langevin method to the resulting complex-weight matrix integral.Time and spatial observables are defined from matrix configurations, with holomorphic time ordering implemented by auxiliary variables, and spatial structure characterized through block observables such as the extent R 2 (t) and the moment-of-inertia tensor.In the bosonic model we find approximately real time and no indication of spontaneous breaking of the spatial symmetry in the i parameter range studied.Including fermions, we control the singular-drift problem induced by the Pfaffian by introducing a fermionic mass regulator m f and an anisotropic mass deformation, which enhances fermionic effects while maintaining numerical stability.In this setup we observe a real, expanding spatial extent and a dynamical dominance of three spatial directions at late times, consistent with a (3 + 1)-dimensional expanding spacetime.We finally discuss the intended limiting procedure-taking the large-N limit first and then approaching the undeformed theory-and comment on future directions, including a comparison with partially supersymmetric deformations that preserve 16 supercharges.

•Mar 24, 2026•0 citations

Quantum Decoherence from Complex Saddle Points

Jun Nishimura, Hiromasa Watanabe

Quantum decoherence is the effect that bridges quantum physics to well-understood classical physics. As such, it plays a crucial role in understanding the mysterious nature of quantum physics. Quantum decoherence is also a source of quantum noise that has to be well under control in quantum computing and in various experiments based on quantum technologies. Here we point out that quantum decoherence can be captured by complex saddle points in the Feynman path integral in much the same way as quantum tunneling can be captured by instantons. In particular, we present some first-principle calculations in the Caldeira-Leggett model, which reproduce the predicted scaling behavior of quantum decoherence with respect to the parameters of the environment such as the temperature and the coupling to the system of interest. We also discuss how to extend our approach to general models by Monte Carlo calculations using a recently developed method to overcome the sign problem.

arXiv:2408.16627•Physical Review Letters•May 27, 2025•3 citations

Real-time quantum dynamics, path integrals and the method of thimbles

Zong-Gang Mou, Paul M. Saffin, Anders Tranberg B +1 more

Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy's theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.

arXiv:1902.09147•Feb 25, 2019•2 citations

Path integral formulation of dissipative quantum dynamics

Alexey Novikov

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

•Jan 20, 2005•0 citations

Real-time quantum dynamics, path integrals and the method of thimbles

Zong-Gang Mou, Paul M. Saffin, Anders Tranberg B +1 more

Abstract Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy’s theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.

Journal of High Energy Physics•Jun 1, 2019•34 citations

Computing real-time quantum path integrals on Sewed, almost-Lefschetz thimbles

Zong-Gang Mou, Paul M. Saffin, Anders Tranberg B

A bstract We present a method to compute real-time path integrals numerically, by Monte-Carlo sampling on near-Lefschetz thimbles. We present a collection of tools based on the Lefschetz thimble methods, which together provide an alternative to existing methods such as the Generalised thimble. These involve a convenient coordinate parameterization of the thimble, direct numerical integration along a radial coordinate into an effective path integral weight and locally deforming the Lefschetz thimble using its Gaussian (non-interacting theory) counterpart in a region about the critical point. We apply this to quantum mechanics, identify possible pitfalls and benefits, and benchmark its efficiency.

Journal of High Energy Physics•Mar 7, 2025•1 citation
A bstract We propose first-principle calculations of an open system based on the real-time path integral formalism treating the environment as well as the system of our interest together on a computer. The sign problem that occurs in applying Monte Carlo methods can be overcome in general by using the so-called Lefschetz thimble method, which has been developed over the past decade. Here we focus on the Caldeira-Leggett model, which is well known, in particular, as a model of quantum decoherence. In this case, the calculation simplifies drastically since the path integral becomes Gaussian for typical initial conditions. The relevant saddle point, which is unique and complex, can be determined by solving a linear equation with a huge but sparse coefficient matrix, and the integration over the Lefschetz thimble can be performed analytically. Thus we obtain, without assumptions or approximations, the reduced density matrix after a long-time evolution, tracing out a large number of harmonic oscillators in the environment. In particular, we confirm the dependence of the decoherence time on the coupling constant and the temperature that has been predicted from the master equation in a certain parameter regime.
Real-time path integrals provide a natural formulation of quantum dynamics also in situations where a Hamiltonian description is unavailable or impractical, but their direct evaluation is obstructed by severe phase cancellation from the oscillatory weight.A guiding theme of this thesis is that, after complexifying the integration variables, real-time path integrals are naturally organized by complex saddle points.In favorable cases the original contour can be replaced by steepest-descent manifolds (Lefschetz thimbles) attached to the relevant saddles, providing a controlled saddle-based description of real-time observables.In Part I, we study quantum decoherence in a closed-system real-time path-integral formulation: the total system (subsystem plus environment) evolves unitarily, and the environment is traced out only at the final time to obtain the reduced density matrix.For the Caldeira-Leggett model with Gaussian initial conditions, the discretized effective action becomes a quadratic form, so that the reduced density matrix is characterized by a complex saddle point and the thimble integral reduces to a finite-dimensional Gaussian integral.The complex saddle is obtained by solving a large but sparse linear system, allowing an exact evaluation of the reduced density matrix without relying on the usual master-equation approximations.We then extend this methodology to a three-dimensional cubic lattice toy model by separating center-of-mass and relative coordinates, treating the center-of-mass mode as the subsystem and the relative modes as an effective environment.This enables controlled numerical studies of how the suppression of off-diagonal matrix elements depends on the coupling, inverse temperature, and system size, and allows us to extract the early-time scaling behavior of decoherence diagnostics from the complex-saddle evaluation.In Part II, we investigate spacetime dynamics in the Lorentzian IKKT matrix model.To define a Lorentzian formulation that is not continuously connected to the Euclidean model by the standard contour deformation, we include a Lorentz invariant mass term with coefficient γ, for which the classical equations admit real, smooth, expanding solutions when γ > 0. Since the Lorentz group is noncompact, the path integral can contain a divergent volume from the boost directions; we remove this volume nonperturbatively by implementing a boost "gauge-fixing" that selects the frame minimizing tr(A 0 ) 2 , leading to the condition tr(A 0 A i ) = 0 with the associated Faddeev-Popov determinant.We then apply the complex Langevin method to the resulting complex-weight matrix integral.Time and spatial observables are defined from matrix configurations, with holomorphic time ordering implemented by auxiliary variables, and spatial structure characterized through block observables such as the extent R 2 (t) and the moment-of-inertia tensor.In the bosonic model we find approximately real time and no indication of spontaneous breaking of the spatial symmetry in the i parameter range studied.Including fermions, we control the singular-drift problem induced by the Pfaffian by introducing a fermionic mass regulator m f and an anisotropic mass deformation, which enhances fermionic effects while maintaining numerical stability.In this setup we observe a real, expanding spatial extent and a dynamical dominance of three spatial directions at late times, consistent with a (3 + 1)-dimensional expanding spacetime.We finally discuss the intended limiting procedure-taking the large-N limit first and then approaching the undeformed theory-and comment on future directions, including a comparison with partially supersymmetric deformations that preserve 16 supercharges.
Quantum decoherence is the effect that bridges quantum physics to well-understood classical physics. As such, it plays a crucial role in understanding the mysterious nature of quantum physics. Quantum decoherence is also a source of quantum noise that has to be well under control in quantum computing and in various experiments based on quantum technologies. Here we point out that quantum decoherence can be captured by complex saddle points in the Feynman path integral in much the same way as quantum tunneling can be captured by instantons. In particular, we present some first-principle calculations in the Caldeira-Leggett model, which reproduce the predicted scaling behavior of quantum decoherence with respect to the parameters of the environment such as the temperature and the coupling to the system of interest. We also discuss how to extend our approach to general models by Monte Carlo calculations using a recently developed method to overcome the sign problem.
Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy's theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
Abstract Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy’s theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.
A bstract We present a method to compute real-time path integrals numerically, by Monte-Carlo sampling on near-Lefschetz thimbles. We present a collection of tools based on the Lefschetz thimble methods, which together provide an alternative to existing methods such as the Generalised thimble. These involve a convenient coordinate parameterization of the thimble, direct numerical integration along a radial coordinate into an effective path integral weight and locally deforming the Lefschetz thimble using its Gaussian (non-interacting theory) counterpart in a region about the critical point. We apply this to quantum mechanics, identify possible pitfalls and benefits, and benchmark its efficiency.