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Physical Review B

Crossover from coherent to incoherent quantum dynamics due to sub-Ohmic dephasing

P. Nalbach, Michael Thorwart

Published Jan 28, 2013• 29 citations
Publisher

Abstract

We report exact results for the influence of purely sub-Ohmic dephasing on the dynamics of a quantum two-level system. From response functions, we determine a crossover coupling strength between oscillatory coherent and overdamped dynamics. Surprisingly, we find no overdamping even at arbitrary large dephasing for spectra with spectral exponent $s<{s}_{c}$. For ${s}_{c}<s\ensuremath{\le}1$ we observe a crossover to overdamped dynamics but a reemergence of oscillatory behavior at very large couplings. Thus experimentally tuning the character of fluctuations from relaxational to dephasing should allow us to suppress the dephasing character of sub-Ohmic and thus $1/f$ noise.

DOI
10.1103/physrevb.87.014116

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We report exact results for the influence of purely sub-Ohmic dephasing on the dynamics of a quantum two-level system. From response functions, we determine a crossover coupling strength between oscillatory coherent and overdamped dynamics. Surprisingly, we find no overdamping even at arbitrary large dephasing for spectra with spectral exponent s<scs<{s}_{c}s<sc​. For sc<s\ensuremath≤1{s}_{c}<s\ensuremath{\le}1sc​<s\ensuremath≤1 we observe a crossover to overdamped dynamics but a reemergence of oscillatory behavior at very large couplings. Thus experimentally tuning the character of fluctuations from relaxational to dephasing should allow us to suppress the dephasing character of sub-Ohmic and thus 1/f1/f1/f 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.
The decoherence of a two-state system coupled with a sub-Ohmic bath is investigated theoretically by means of the perturbation approach based on a unitary transformation. It is shown that the decoherence depends strongly and sensitively on the structure of environment. Nonadiabatic effect is treated through the introduction of a function ${\ensuremath{\xi}}_{k}$ which depends on the boson frequency and renormalized tunneling. The results are as follows: (i) the nonequilibrium correlation function P(t)P(t)P(t), the dynamical susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$, and the equilibrium correlation function C(t)C(t)C(t) are analytically obtained for $s\ensuremath{\leqslant}1$; (ii) the phase diagram of thermodynamic transition shows the delocalized-localized transition point ${\ensuremath{\alpha}}_{l}$ which agrees with exact results and numerical data from the numerical renormalization group; (iii) the dynamical transition point ${\ensuremath{\alpha}}_{c}$ between coherent and incoherent phase is explicitly given. A crossover from the coherent oscillation to incoherent relaxation appears with increasing coupling (for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the coherent dynamics disappear); (iv) the Shiba's relation and sum rule are exactly satisfied when $\ensuremath{\alpha}\ensuremath{\leqslant}{\ensuremath{\alpha}}_{c}$; (v) an underdamping-overdamping transition point ${\ensuremath{\alpha}}_{c}^{*}$ exists in the function $S(\ensuremath{\omega})$. Consequently, the dynamical phase diagrams in both ohmic and sub-Ohmic case are mapped out. For $\ensuremath{\Delta}⪡{\ensuremath{\omega}}_{c}$, the critical couplings (${\ensuremath{\alpha}}_{l},{\ensuremath{\alpha}}_{c}$, and ${\ensuremath{\alpha}}_{c}^{*}$) are proportional to ${\ensuremath{\Delta}}^{1\ensuremath{-}s}$.
We study a quantum two-level system under the influence of two independent baths, i.e., a sub-Ohmic pure dephasing bath and an Ohmic or sub-Ohmic relaxational bath. We show that cooling such a system invariably polarizes one of the two baths. A polarized relaxational bath creates an effective asymmetry. This asymmetry can be suppressed by additional dephasing noise. This being less effective, the more dominant low frequencies are in the dephasing noise. A polarized dephasing bath generates a large shift in the coherent oscillation frequency of the two-level system. This frequency shift is little affected by additional relaxational noise nor by the frequency distribution of the dephasing noise itself. As our model reflects a typical situation for superconducting phase qubits, our findings can help optimize cooling protocols for future quantum electronic devices.
We study the nonequilibrium dynamics of a quantum system under the influence of two noncommuting fluctuation sources, i.e., purely dephasing fluctuations and relaxational fluctuations. We find that increasing purely dephasing fluctuations suppress increasing relaxation in the quantum system. This effect is further enhanced when both fluctuation sources are fully correlated. These effects arise for medium to strong primary fluctuations already when the secondary fluctuations are weak due to their noncommuting coupling to the quantum system. Dephasing, in contrast, is increased by increasing any of the two fluctuations. Fully correlated fluctuations result in overdamping at much lower system-bath coupling than uncorrelated noncommuting fluctuations. In total, we observe that treating subdominant secondary environmental fluctuations perturbatively leads, as neglecting them, to erroneous conclusions.
We study the dissipative quantum dynamics and the asymptotic behavior of a particle in a bistable potential interacting with a sub-Ohmic broadband environment. The reduced dynamics, in the intermediate to strong dissipation regime, is obtained beyond the two-level system approximation by using a real-time path integral approach. We find a crossover dynamic regime with damped intra-well oscillations and incoherent tunneling and a completely incoherent regime at strong damping. Moreover, a nonmonotonic behavior of the left/right well population difference is found as a function of the damping strength.
We investigate the dynamical quantum Fisher information of a two-level system coupled to a bosonic environment, focusing on the estimation of the qubit gap. We combine analytical calculations with numerically controlled matrix-product-state simulations. In the exactly solvable pure-dephasing Ohmic regime at zero temperature, the long-time quantum Fisher information displays a coupling-dependent algebraic behavior, leading to a dynamical crossover: it grows without bound at weak coupling, approaches a finite asymptotic value at the crossover coupling, and vanishes at strong coupling. At finite temperature, thermal fluctuations suppress the long-time growth and generate a finite-time maximum, whose dependence on the dephasing coupling retains a clear signa- ture of the zero-temperature crossover. We show that this crossover is absent at zero temperature in non-Ohmic baths: the long-time quantum Fisher information vanishes in the sub-Ohmic and diverges in the super-Ohmic regimes. At non-zero temperature instead, the crossover appears for a super-Ohmic quadratic bath and the crossover coupling becomes temperature-dependent. Moreover, in the zero-temperature Ohmic regime, we introduce an additional amplitude-damping system-bath coupling that induces energy relaxation. This relaxation channel replaces the unbounded long-time growth with a finite asymptotic quantum Fisher information associated with the reduced interacting ground state, while a signature of the pure-dephasing crossover persists in the early-time dynamics. These results establish a direct connection between the low-frequency structure of the bath and the asymptotic metrological behavior of dynamical gap sensing, and show how thermal fluctuations and energy relaxation regularize the ideal pure-dephasing dynamical crossover of the quantum Fisher information.