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arXiv:2208.10313 · Physical Review Letters

Hidden Phase of the Spin-Boson Model

Florian Otterpohl, P. Nalbach, Michael Thorwart

Published Sep 16, 2022• 35 citations
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Abstract

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.

DOI
10.1103/physrevlett.129.120406

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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.
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.
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.
Dephasing in quantum systems is typically the result of their interaction with environmental degrees of freedom. We investigate within a spin-boson model the influence of a super-Ohmic environment on the dynamics of a quantum two-state system. A super-Ohmic environment thereby models typical bulk phonons which are a common disturbance for solid state quantum systems as, for example, nitrogen-vacancy centers. By applying the numerically exact quasiadiabatic path-integral approach we show that for strong system-bath coupling, pseudocoherent dynamics emerges, i.e., oscillatory dynamics at short times due to slaving of the quantum system to the bath dynamics. We extend the phase diagram known for sub-Ohmic and Ohmic environments into the super-Ohmic regime and observe a pronounced nonmonotonous behavior. Super-Ohmic purely dephasing fluctuations strongly suppress the amplitude of coherent dynamics at very short times with no subsequent further decay at later times. Nevertheless, they render the dynamics overdamped. The corresponding phase separation line shows also a nonmonotonous behavior, very similar to the pseudocoherent dynamics.
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 analyze the static and dynamic properties of two Ising-coupled quantum spins embedded in a common bosonic bath as an archetype of dissipative quantum mechanics. First, we elucidate the ground-state phase diagram for an Ohmic and a sub-Ohmic bath using a combination of bosonic numerical renormalization group (NRG), analytical techniques, and intuitive arguments. Second, by employing the time-dependent NRG we investigate the system's rich dynamical behavior arising from the complex interplay between spin-spin and spin-bath interactions. Interestingly, spin oscillations can synchronize due to the proximity of the common non-Markovian bath and the system displays highly entangled steady states for certain nonequilibrium initial preparations. We complement our nonperturbative numerical results by exact analytical solutions when available and provide quantitative limits on the applicability of the perturbative Bloch-Redfield approach at weak coupling.
The spin-boson model, describing a two-level system coupled to a bosonic bath, is extensively studied as a paradigmatic dissipative quantum system, exhibiting rich dynamical behavior and even a localization transition in the strong coupling regime. Here, we additionally consider dephasing as a source of Markovian dissipation on top of the non-Markovian dynamics due to an Ohmic bath, and investigate the dynamics of the spin. We show that the characteristic frequency of the spin dynamics, while strongly renormalized by the bosonic bath, changes in a simple fashion (or does not change at all) with dephasing. To obtain these results, we develop an exact nonperturbative method known as the stochastic Schr\"odinger equation, mimicking the Ohmic bath via a stochastic magnetic field combined with the Lindblad quantum master equation due to dephasing, which allows us to numerically compute the dynamics. Furthermore, we derive weak-coupling analytic results utilizing the well-known noninteracting blip approximation. Our findings are relevant to quantum simulation of the spin-boson model in the regime of strong coupling in trapped ions and circuit QED architectures among others.