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:0908.2749 · Physical Review B

Ultraslow quantum dynamics in a sub-Ohmic heat bath

P. Nalbach, Michael Thorwart

Published Feb 25, 2010• 70 citations
arXivHTMLPDF

Abstract

We show that the low-frequency modes of a sub-Ohmic bosonic heat bath generate an effective dynamical asymmetry for an intrinsically symmetric quantum spin-1/2. An initially fully polarized spin first decays toward a quasiequilibrium determined by the dynamical asymmetry, thereby showing coherent damped oscillations on the (fast) time scale of the spin splitting. On top of this, the dynamical asymmetry itself decays on an ultraslow time scale and vanishes asymptotically since the global equilibrium phase is symmetric. We quantitatively study the nature of the initial fast decay to the quasiequilibrium and discuss the features of ultraslow dynamics of the quasiequilibrium itself. The dynamical asymmetry is more pronounced for smaller values of the sub-Ohmic exponent and for lower temperatures, which emphasizes the quantum many-body nature of the effect. The symmetry breaking is related to the dynamic crossover between coherent and overdamped relaxation of the spin polarization and is not connected to the localization quantum phase transition. In addition to this delocalized phase, we identify a phase which is characterized by damped coherent oscillations in the localized phase. This allows for a sketch of the zero-temperature phase diagram of the sub-Ohmic spin-boson model with four distinct phases.

DOI
10.1103/physrevb.81.054308

Related papers

Transient Dynamical Phase Diagram of the Spin-Boson Model

Olga Goulko, Hsing-Ta Chen, Moshe Goldstein +1 more

We investigate the real-time dynamics of the sub-Ohmic spin-boson model across a broad range of coupling strengths, using the numerically exact inchworm quantum Monte Carlo algorithm. From short- and intermediate-time dynamics starting from an initially decoupled state, we extract signatures of the zero-temperature quantum phase transition between localized and delocalized states. We show that the dynamical phase diagram thus obtained differs from the equilibrium phase diagram in both the values of critical couplings and the associated values of the critical exponent. We also identify and quantitatively analyze two competing mechanisms for the crossover between coherent oscillations and incoherent decay. Deep in the sub-Ohmic regime, the crossover is driven by the damping of the oscillation amplitude, while closer to the Ohmic regime the oscillation frequency itself drops sharply to zero at large coupling.

arXiv:2402.18561•Physical Review Letters•Feb 6, 2025•8 citations

Transient dynamical phase diagram of the spin-boson model at finite temperature

Olga Goulko, Hsing-Ta Chen, Moshe Goldstein +1 more

We present numerically exact inchworm quantum Monte Carlo results for the real-time dynamics of the spin polarization in the sub-Ohmic spin-boson model at finite temperature. We focus in particular on the localization and coherence behavior of the model, extending our previous study at low temperature [Phys. Rev. Lett. 134, 056502 (2025)]. As the temperature increases, the system becomes less localized and less coherent. The loss of coherence, which is controlled by two independent mechanisms -- a smooth damping-driven crossover and a sharp frequency-driven transition -- exhibits a nontrivial temperature dependence. While both types of coherence loss occur at lower coupling in the high temperature regime, the frequency exhibits a sharper drop at high temperatures and this drop is observed for all values of the sub-Ohmic exponent, in contrast to the zero-temperature case. We discuss the full temperature-dependent dynamical phase diagram of the system and the interplay between coherence and localization across a wide range of physical parameters.

arXiv:2509.02345•Physical review. B./Physical review. B•Mar 5, 2026•1 citation

Equilibrium and Nonequilibrium Dynamics of the Sub-Ohmic Spin-Boson Model

Frithjof B. Anders, R. Bulla, Matthias Vojta C

Employing the nonperturbative numerical renormalization group method, we study the dynamics of the spin-boson model, which describes a two-level system coupled to a bosonic bath with a spectral density J(omega) proportional to omega(s). We show that, in contrast with the case of Ohmic damping, the delocalized phase of the sub-Ohmic model cannot be characterized by a single energy scale only, due to the presence of a nontrivial quantum phase transition. In the strongly sub-Ohmic regime, s<<1, weakly damped coherent oscillations on short time scales are possible even in the localized phase--this is of crucial relevance, e.g., for qubits subject to electromagnetic noise.

arXiv:cond-mat/0607443•Physical Review Letters•May 24, 2007•138 citations

Hidden Phase of the Spin-Boson Model

Florian Otterpohl, P. Nalbach, Michael Thorwart

A quantum two-level system immersed in a sub-Ohmic bath experiences enhanced low-frequency quantum statistical fluctuations which render the nonequilibrium quantum dynamics highly non-Markovian. Upon using the numerically exact time-evolving matrix product operator approach, we investigate the phase diagram of the polarization dynamics. In addition to the known phases of damped coherent oscillatory dynamics and overdamped decay, we identify a new third region in the phase diagram for strong coupling showing an aperiodic behavior. We determine the corresponding phase boundaries. The dynamics of the quantum two-state system herein is not coherent by itself but slaved to the oscillatory bath dynamics.

arXiv:2208.10313•Physical Review Letters•Sep 16, 2022•35 citations

Dynamics of the sub-Ohmic spin-boson model: A time-dependent variational study

Ning Wu, Liwei Duan, Xin Li +1 more

The Dirac-Frenkel time-dependent variation is employed to probe the dynamics of the zero temperature sub-Ohmic spin-boson model with strong friction utilizing the Davydov D1 ansatz. It is shown that initial conditions of the phonon bath have considerable influence on the dynamics. Counterintuitively, even in the very strong coupling regime, quantum coherence features still manage to survive under the polarized bath initial condition, while such features are absent under the factorized bath initial condition. In addition, a coherent-incoherent transition is found at a critical coupling strength α ≈ 0.1 for s = 0.25 under the factorized bath initial condition. We quantify how faithfully our ansatz follows the Schrödinger equation, finding that the time-dependent variational approach is robust for strong dissipation and deep sub-Ohmic baths (s ≪ 1).

arXiv:1302.1682•The Journal of Chemical Physics•Feb 28, 2013•42 citations

Quantum dynamics simulations in an ultraslow bath using hierarchy of stochastic Schrödinger equations

Yaling Ke, Yi Zhao

The hierarchy of stochastic Schrödinger equation, previously developed under the unpolarised initial bath states, is extended in this paper for open quantum dynamics under polarised initial bath conditions. The method is proved to be a powerful tool in investigating quantum dynamics exposed to an ultraslow Ohmic bath, as in this case the hierarchical truncation level and the random sampling number can be kept at a relatively small extent. By systematically increasing the system–bath coupling strength, the symmetric Ohmic spin-boson dynamics is investigated at finite temperature, with a very small cut-off frequency. It is confirmed that the slow bath makes the system dynamics extremely sensitive to the initial bath conditions. The localisation tendency is stronger in the polarised initial bath conditions. Besides, the oscillatory coherent dynamics persists even when the system–bath coupling is very strong, in correspondence with what is found recently in the deep sub-Ohmic bath, where also the low-frequency modes dominate.

Molecular Physics•Jan 30, 2018•5 citations
We show that the low-frequency modes of a sub-Ohmic bosonic heat bath generate an effective dynamical asymmetry for an intrinsically symmetric quantum spin-1/2. An initially fully polarized spin first decays toward a quasiequilibrium determined by the dynamical asymmetry, thereby showing coherent damped oscillations on the (fast) time scale of the spin splitting. On top of this, the dynamical asymmetry itself decays on an ultraslow time scale and vanishes asymptotically since the global equilibrium phase is symmetric. We quantitatively study the nature of the initial fast decay to the quasiequilibrium and discuss the features of ultraslow dynamics of the quasiequilibrium itself. The dynamical asymmetry is more pronounced for smaller values of the sub-Ohmic exponent and for lower temperatures, which emphasizes the quantum many-body nature of the effect. The symmetry breaking is related to the dynamic crossover between coherent and overdamped relaxation of the spin polarization and is not connected to the localization quantum phase transition. In addition to this delocalized phase, we identify a phase which is characterized by damped coherent oscillations in the localized phase. This allows for a sketch of the zero-temperature phase diagram of the sub-Ohmic spin-boson model with four distinct phases.
We investigate the real-time dynamics of the sub-Ohmic spin-boson model across a broad range of coupling strengths, using the numerically exact inchworm quantum Monte Carlo algorithm. From short- and intermediate-time dynamics starting from an initially decoupled state, we extract signatures of the zero-temperature quantum phase transition between localized and delocalized states. We show that the dynamical phase diagram thus obtained differs from the equilibrium phase diagram in both the values of critical couplings and the associated values of the critical exponent. We also identify and quantitatively analyze two competing mechanisms for the crossover between coherent oscillations and incoherent decay. Deep in the sub-Ohmic regime, the crossover is driven by the damping of the oscillation amplitude, while closer to the Ohmic regime the oscillation frequency itself drops sharply to zero at large coupling.
We present numerically exact inchworm quantum Monte Carlo results for the real-time dynamics of the spin polarization in the sub-Ohmic spin-boson model at finite temperature. We focus in particular on the localization and coherence behavior of the model, extending our previous study at low temperature [Phys. Rev. Lett. 134, 056502 (2025)]. As the temperature increases, the system becomes less localized and less coherent. The loss of coherence, which is controlled by two independent mechanisms -- a smooth damping-driven crossover and a sharp frequency-driven transition -- exhibits a nontrivial temperature dependence. While both types of coherence loss occur at lower coupling in the high temperature regime, the frequency exhibits a sharper drop at high temperatures and this drop is observed for all values of the sub-Ohmic exponent, in contrast to the zero-temperature case. We discuss the full temperature-dependent dynamical phase diagram of the system and the interplay between coherence and localization across a wide range of physical parameters.
Employing the nonperturbative numerical renormalization group method, we study the dynamics of the spin-boson model, which describes a two-level system coupled to a bosonic bath with a spectral density J(omega) proportional to omega(s). We show that, in contrast with the case of Ohmic damping, the delocalized phase of the sub-Ohmic model cannot be characterized by a single energy scale only, due to the presence of a nontrivial quantum phase transition. In the strongly sub-Ohmic regime, s<<1, weakly damped coherent oscillations on short time scales are possible even in the localized phase--this is of crucial relevance, e.g., for qubits subject to electromagnetic noise.
A quantum two-level system immersed in a sub-Ohmic bath experiences enhanced low-frequency quantum statistical fluctuations which render the nonequilibrium quantum dynamics highly non-Markovian. Upon using the numerically exact time-evolving matrix product operator approach, we investigate the phase diagram of the polarization dynamics. In addition to the known phases of damped coherent oscillatory dynamics and overdamped decay, we identify a new third region in the phase diagram for strong coupling showing an aperiodic behavior. We determine the corresponding phase boundaries. The dynamics of the quantum two-state system herein is not coherent by itself but slaved to the oscillatory bath dynamics.
The Dirac-Frenkel time-dependent variation is employed to probe the dynamics of the zero temperature sub-Ohmic spin-boson model with strong friction utilizing the Davydov D1 ansatz. It is shown that initial conditions of the phonon bath have considerable influence on the dynamics. Counterintuitively, even in the very strong coupling regime, quantum coherence features still manage to survive under the polarized bath initial condition, while such features are absent under the factorized bath initial condition. In addition, a coherent-incoherent transition is found at a critical coupling strength α ≈ 0.1 for s = 0.25 under the factorized bath initial condition. We quantify how faithfully our ansatz follows the Schrödinger equation, finding that the time-dependent variational approach is robust for strong dissipation and deep sub-Ohmic baths (s ≪ 1).
The hierarchy of stochastic Schrödinger equation, previously developed under the unpolarised initial bath states, is extended in this paper for open quantum dynamics under polarised initial bath conditions. The method is proved to be a powerful tool in investigating quantum dynamics exposed to an ultraslow Ohmic bath, as in this case the hierarchical truncation level and the random sampling number can be kept at a relatively small extent. By systematically increasing the system–bath coupling strength, the symmetric Ohmic spin-boson dynamics is investigated at finite temperature, with a very small cut-off frequency. It is confirmed that the slow bath makes the system dynamics extremely sensitive to the initial bath conditions. The localisation tendency is stronger in the polarised initial bath conditions. Besides, the oscillatory coherent dynamics persists even when the system–bath coupling is very strong, in correspondence with what is found recently in the deep sub-Ohmic bath, where also the low-frequency modes dominate.