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Hole spin qubits based on germanium (Ge) have strong tunable spin orbit interaction (SOI) and ultrafast qubit operation speed. Here we report that the Rabi frequency (f_Rabi) of a hole spin qubit in a Ge hut wire (HW) double quantum dot (DQD) is electrically tuned through the detuning energy and middle gate voltage (V_M). f_Rabi gradually decreases with increasing detuning energy; on the contrary, f_Rabi is positively correlated with V_M. We attribute our results to the change of electric field on SOI and the contribution of the excited state in quantum dots to f_Rabi. We further demonstrate an ultrafast f_Rabi exceeding 1.2 GHz, which evidences the strong SOI in our device. The discovery of an ultrafast and electrically tunable f_Rabi in a hole spin qubit has potential applications in semiconductor quantum computing., Comment: 19 pages, 4 figures

Quantum computers have made extraordinary progress over the past decade, and significant milestones have been achieved along the path of pursuing universal fault-tolerant quantum computers. Quantum advantage, the tipping point heralding the quantum era, has been accomplished along with several waves of breakthroughs. Quantum hardware has become more integrated and architectural compared to its toddler days. The controlling precision of various physical systems is pushed beyond the fault-tolerant threshold. Meanwhile, quantum computation research has established a new norm by embracing industrialization and commercialization. The joint power of governments, private investors, and tech companies has significantly shaped a new vibrant environment that accelerates the development of this field, now at the beginning of the noisy intermediate-scale quantum era. Here, we first discuss the progress achieved in the field of quantum computation by reviewing the most important algorithms and advances in the most promising technical routes, and then summarizing the next-stage challenges. Furthermore, we illustrate our confidence that solid foundations have been built for the fault-tolerant quantum computer and our optimism that the emergence of quantum killer applications essential for human society shall happen in the future.

The coexistence of superconductivity and topology holds the potential to realize exotic quantum states of matter. Here we report that superconductivity induced by high pressure in three thallium-based materials, covering the phase transition from a normal insulator (TlBiS2) to a topological insulator (TlBiSe2) through a Dirac semimetal (TlBiSeS). By increasing the pressure up to 60 GPa, we observe superconductivity phase diagrams with maximal Tc values at 6.0-8.1 K. Our density-functional theory calculations reveal topological surface states in superconductivity phases for all three compounds. Our study paves the path to explore topological superconductivity and topological phase transitions., Comment: 39 pages, 5+22 figures

Long coherence time and compatibility with semiconductor fabrication make spin qubits in silicon an attractive platform for quantum computing. In recent years, hole spin qubits are being developed as they have the advantages of weak coupling to nuclear spin noise and strong spin-orbit coupling (SOC), in constructing high-fidelity quantum gates. However, there are relatively few studies on the hole spin qubits in a single acceptor, which requires only low density of the metallic gates. In particular, the investigation of flexible tunability using controllable strain for fault-tolerant quantum gates of acceptor-based qubits is still lacking. Here, we study the tunability of electric dipole spin resonance (EDSR) of acceptor-based hole spin qubits with controllable strain. The flexible tunability of LH-HH splitting and spin-hole coupling (SHC) with the two kinds of strain can avoid high electric field at the "sweet spot", and the operation performance of the acceptor qubits could be optimized. Longer relaxation time or stronger EDSR coupling at low electric field can be obtained. Moreover, with asymmetric strain, two "sweet spots" are induced and may merge together, and form a second-order "sweet spot". As a result, the quality factor $Q$ can reach $10^{4}$ for single-qubit operation, with high tolerance for the electric field variation. Furthermore, the two-qubit operation of acceptor qubits based on dipole-dipole interaction is discussed for high-fidelity two-qubit gates. The tunability of spin qubit properties in acceptor via strain could provide promising routes for spin-based quantum computing., Comment: 17 pages, 12 figures

Real-word search and recommender systems usually adopt a multi-stage ranking architecture, including matching, pre-ranking, ranking, and re-ranking. Previous works mainly focus on the ranking stage while very few focus on the pre-ranking stage. In this paper, we focus on the information transfer from ranking to pre-ranking stage. We propose a new Contrastive Information Transfer (CIT) framework to transfer useful information from ranking model to pre-ranking model. We train the pre-ranking model to distinguish the positive pair of representation from a set of positive and negative pairs with a contrastive objective. As a consequence, the pre-ranking model can make full use of rich information in ranking model's representations. The CIT framework also has the advantage of alleviating selection bias and improving the performance of recall metrics, which is crucial for pre-ranking models. We conduct extensive experiments including offline datasets and online A/B testing. Experimental results show that CIT achieves superior results than competitive models. In addition, a strict online A/B testing at one of the world's largest E-commercial platforms shows that the proposed model achieves 0.63\% improvements on CTR and 1.64\% improvements on VBR. The proposed model now has been deployed online and serves the main traffic of this system, contributing a remarkable business growth., Comment: 5 pages

Rich user behavior data has been proven to be of great value for Click-Through Rate (CTR) prediction applications, especially in industrial recommender, search, or advertising systems. However, it's non-trivial for real-world systems to make full use of long-term user behaviors due to the strict requirements of online serving time. Most previous works adopt the retrieval-based strategy, where a small number of user behaviors are retrieved first for subsequent attention. However, the retrieval-based methods are sub-optimal and would cause more or less information losses, and it's difficult to balance the effectiveness and efficiency of the retrieval algorithm. In this paper, we propose SDIM (Sampling-based Deep Interest Modeling), a simple yet effective sampling-based end-to-end approach for modeling long-term user behaviors. We sample from multiple hash functions to generate hash signatures of the candidate item and each item in the user behavior sequence, and obtain the user interest by directly gathering behavior items associated with the candidate item with the same hash signature. We show theoretically and experimentally that the proposed method performs on par with standard attention-based models on modeling long-term user behaviors, while being sizable times faster. We also introduce the deployment of SDIM in our system. Specifically, we decouple the behavior sequence hashing, which is the most time-consuming part, from the CTR model by designing a separate module named BSE (behavior Sequence Encoding). BSE is latency-free for the CTR server, enabling us to model extremely long user behaviors. Both offline and online experiments are conducted to demonstrate the effectiveness of SDIM. SDIM now has been deployed online in the search system of Meituan APP., Comment: Accepted by CIKM 2022, 10 pages

Industrial search and recommendation systems mostly follow the classic multi-stage information retrieval paradigm: matching, pre-ranking, ranking, and re-ranking stages. To account for system efficiency, simple vector-product based models are commonly deployed in the pre-ranking stage. Recent works consider distilling the high knowledge of large ranking models to small pre-ranking models for better effectiveness. However, two major challenges in pre-ranking system still exist: (i) without explicitly modeling the performance gain versus computation cost, the predefined latency constraint in the pre-ranking stage inevitably leads to suboptimal solutions; (ii) transferring the ranking teacher's knowledge to a pre-ranking student with a predetermined handcrafted architecture still suffers from the loss of model performance. In this work, a novel framework AutoFAS is proposed which jointly optimizes the efficiency and effectiveness of the pre-ranking model: (i) AutoFAS for the first time simultaneously selects the most valuable features and network architectures using Neural Architecture Search (NAS) technique; (ii) equipped with ranking model guided reward during NAS procedure, AutoFAS can select the best pre-ranking architecture for a given ranking teacher without any computation overhead. Experimental results in our real world search system show AutoFAS consistently outperforms the previous state-of-the-art (SOTA) approaches at a lower computing cost. Notably, our model has been adopted in the pre-ranking module in the search system of Meituan, bringing significant improvements.

Superexchange is one of the vital resources to realize long-range interaction between distant spins for large-scale quantum computing. Recent experiments have demonstrated coherent oscillations between logical states defined by remote spins whose coupling is given by the superexchange interaction mediated by central spins. Excavating the potential of superexchange requires a full understanding of the interaction in terms of control parameters, which is still lacking in literature. Here, using full configuration interaction calculations, we study a two-electron system in a linear triple-quantum-dot device in which the left and right dots are occupied by a single electron each, whose spin states are defined as qubits. The numerical nature of the full configuration interaction calculations allows access to the microscopic details of the quantum-dot confining potential and electronic wavefunctions, some of which are overlooked in the celebrated Hubbard model but turn out to be critical for the behavior of superexchange. We focus on the detuning regime where the charge ground state yields an empty middle dot. We have found that, when the detunings at the left and right dots are leveled, the superexchange can exhibit a non-monotonic behavior which ranges from positive to negative values as a function of the middle-dot detuning. We further show that a larger relative detuning between the left and right dots causes the magnitude of the superexchange to increase (decrease) for an originally positive (negative) superexchange. We then proceed to show the results for a much larger left-right dot detuning. Our results suggest that even a simple configuration of delocalized two-electron states in a linear triple-quantum-dot device exhibits superexchange energy with non-trivial behaviors, which could have important applications in spin-based quantum computing., Comment: 9+6 pages, 6+1 figures

A spin qubit in semiconductor quantum dots holds promise for quantum information processing for scalability and long coherence time. An important semiconductor qubit system is a double quantum dot trapping two electrons or holes, whose spin states encode either a singlet-triplet qubit or two single-spin qubits coupled by exchange interaction. In this article, we report progress on spin dephasing of two exchange-coupled spins in a double quantum dot. We first discuss the schemes of two-qubit gates and qubit encodings in gate-defined quantum dots or donor atoms based on the exchange interaction. Then, we report the progress on spin dephasing of a singlet-triplet qubit or a two-qubit gate. The methods of suppressing spin dephasing are further discussed. The understanding of spin dephasing may provide insights into the realization of high-fidelity quantum gates for spin-based quantum computing., Comment: Accepted by Advanced Quantum Technologies

Electron spin qubit in a quantum dot has been studied extensively for scalable quantum information processing over the past two decades. Recently, high-fidelity and fast single-spin control and strong spin-photon coupling have been demonstrated using a synthetic spin-orbit coupling created by a micromagnet. Such strong electrical driving suggests a strongly coupled spin-photon system. Here we study the relaxation, pure dephasing, and electrical manipulation of a dressed-spin qubit based on a driven single electron in a quantum dot. We find that the pure dephasing of a dressed qubit due to charge noise can be suppressed substantially at a sweet spot of the dressed qubit. The relaxation at low magnetic fields exhibits non-monotonic behavior due to the energy compensation from the driving field. Moreover, the longitudinal component of the synthetic spin-orbit field could provide fast electric-dipole-induced dressed-spin resonance (EDDSR). Based on the slow dephasing and fast EDDSR, we further propose a scheme for dressed-spin-based semiconductor quantum computing., Comment: arXiv admin note: text overlap with arXiv:2008.04671

Recently monolayer jacutingaite (Pt2HgSe3), a naturally occurring exfoliable mineral, discovered in Brazil in 2008, has been theoretically predicted as a candidate quantum spin Hall system with a 0.5 eV band gap, while the bulk form is one of only a few known dual-topological insulators which may host different surface states protected by symmetries. In this work, we systematically investigate both structure and electronic evolution of bulk Pt2HgSe3 under high pressure up to 96 GPa. The nontrivial topology persists up to the structural phase transition observed in the high-pressure regime. Interestingly, we found that this phase transition is accompanied by the appearance of superconductivity at around 55 GPa and the critical transition temperature Tc increases with applied pressure. Our results demonstrate that Pt2HgSe3 with nontrivial topology of electronic states displays new ground states upon compression and raises potentials in application to the next-generation spintronic devices., Comment: 17 pages,6 figures, 1 table

Tunable synthetic spin-orbit coupling (s-SOC) is one of the key challenges in various quantum systems, such as ultracold atomic gases, topological superconductors, and semiconductor quantum dots. Here we experimentally demonstrate controlling the s-SOC by investigating the anisotropy of spin-valley resonance in a silicon quantum dot. As we rotate the applied magnetic field in-plane, we find a striking nonsinusoidal behavior of resonance amplitude that distinguishes s-SOC from the intrinsic spin-orbit coupling (i-SOC), and associate this behavior with the previously overlooked in-plane transverse magnetic field gradient. Moreover, by theoretically analyzing the experimentally measured s-SOC field, we predict the quality factor of the spin qubit could be optimized if the orientation of the in-plane magnetic field is rotated away from the traditional working point., Comment: 26 pages, 10 figures

An electron spin qubit in silicon quantum dots holds promise for quantum information processing due to the scalability and long coherence. An essential ingredient to recent progress is the employment of micromagnets. They generate a synthetic spin-orbit coupling (SOC), which allows high-fidelity spin manipulation and strong interaction between an electron spin and cavity photons. To scaled-up quantum computing, multiple technical challenges remain to be overcome, including controlling the valley degree of freedom, which is usually considered detrimental to a spin qubit. Here, we show that it is possible to significantly enhance the electrical manipulation of a spin qubit through the effect of constructive interference and the large spin-valley mixing. To characterize the quality of spin control, we also studied spin dephasing due to charge noise through spin-valley mixing. The competition between the increased control strength and spin dephasing produces two sweet-spots, where the quality factor of the spin qubit can be high. Finally, we reveal that the synthetic SOC leads to distinctive spin relaxation in silicon, which explains recent experiments., Comment: Accepted by NPJ Quantum Inf

The electrical control of a spin qubit in a quantum dot relies on spin-orbit coupling (SOC), which could be either intrinsic to the underlying crystal lattice or heterostructure, or extrinsic via, for example, a micro-magnet. Here we show that a key difference between the intrinsic SOC and the synthetic SOC introduced by a micro-magnet is their symmetry under time reversal. Specifically, the time-reversal symmetry ($\mathcal{T}$-symmetry) of the intrinsic SOC leads to not only the traditional van Vleck cancellation known for spin relaxation, but also vanishing spin dephasing to the lowest order of SOC, which we term as "longitudinal spin-orbit field cancellation". On the other hand, the synthetic SOC from a micro-magnet breaks the $\mathcal{T}$-symmetry, therefore eliminates both the "van Vleck cancellation" and the "longitudinal spin-orbit field cancellation". In other words, the effective field $\vec\Omega$ experienced by the spin qubit does not depend on the quantization magnetic field anymore, and a longitudinal component is allowed for $\vec\Omega$ to the first order of SOC. Consequently, spin relaxation and dephasing are qualitatively modified compared with the case of the intrinsic SOC. Furthermore, the fidelity of electric-dipole spin resonance based on $\vec\Omega$ could be optimized, with potential applications in spin-based quantum computing.

The Solar System harbors deep unresolved mysteries despite centuries-long study. A highly intriguing case concerns anomalous non-dipolar and non-axisymmetric magnetic fields of Uranus and Neptune that have long eluded explanation by the prevailing theory. A thin-shell dynamo conjecture captures observed phenomena but leaves unexplained fundamental material basis and underlying mechanism. Here, we report the discovery of trihydrogen oxide (H3O) in metallic liquid state stabilized at extreme pressure and temperature conditions inside these icy planets. Calculated stability pressure field compared to known pressure-radius relation for Uranus and Neptune places metallic liquid H3O in a thin-shell zone near planetary cores. These findings from accurate quantum mechanical calculations rationalize the empirically conjectured thin-shell dynamo model and establish key physical benchmarks that are essential to elucidating the enigmatic magnetic-field anomaly of Uranus and Neptune, resolving a major mystery in planetary science.

Rapid progress in semiconductor spin qubits has enabled experimental demonstrations of a two-qubit logic gate. Understanding spin decoherence in a two-qubit logic gate is necessary for optimal qubit operation. We study spin decoherence due to $1/f$ charge noise for two electrons in a double quantum dot used for a two-qubit controlled-phase gate. In contrast to the usual belief, spin decoherence can be dominated by the tunneling noise from $1/f$ charge noise instead of the detuning noise. Tunneling noise can dominate because the effect of tunneling noise on the spin qubit is first order in the charge admixture; while the effect of the detuning noise is only second order. The different orders of contributions result in different detuning dependence of the decoherence, which provides a way to identify the noise source. We find that decoherence in a recent two-qubit experiment was dominated by the tunneling noise from $1/f$ charge noise. The results illustrate the importance of considering tunneling noise to design optimal operation of spin qubits.

An electron spin qubit in a silicon donor atom is a promising candidate for quantum information processing because of its long coherence time. To be sensed with a single-electron transistor, the donor atom is usually located near an interface, where the donor states can be coupled with interface states. Here we study the phonon-assisted spin-relaxation mechanisms when a donor is coupled to confined (quantum-dot-like) interface states. We find that both Zeeman interaction and spin-orbit interaction can hybridize spin and orbital states, each contributing to phonon-assisted spin relaxation in addition to the spin relaxation for a bulk donor or a quantum dot. When the applied magnetic field $B$ is weak (compared to orbital spacing), the phonon assisted spin relaxation shows the $B^5$ dependence. We find that there are peaks (hot-spots) in the $B$-dependent and detuning dependent spin relaxation due to strong hybridization of orbital states with opposite spin. We also find spin relaxation dips (cool-spots) due to the interference of different relaxation channels. Qubit operations near spin relaxation hot-spots can be useful for the fast spin initialization and near cool-spots for the preservation of quantum information during the transfer of spin qubits.

We report on the challenges and limitations of direct coupling of the magnetic field from a circuit resonator to an electron spin bound to a donor potential. We propose a device consisting of a trilayer lumped-element superconducting resonator and a single donor implanted in enriched $^{28}$Si. The resonator impedance is significantly smaller than the practically achievable limit using prevalent coplanar resonators. Furthermore, the resonator includes a nano-scale spiral inductor to spatially focus the magnetic field from the photons at the location of the implanted donor. The design promises approximately two orders of magnitude increase in the local magnetic field, and thus the spin to photon coupling rate $g$, compared to the estimated coupling rate to the magnetic field of coplanar transmission-line resonators. We show that by using niobium (aluminum) as the resonator's superconductor and a single phosphorous (bismuth) atom as the donor, a coupling rate of $g/2\pi$=0.24 MHz (0.39 MHz) can be achieved in the single photon regime. For this hybrid cavity quantum electrodynamic system, such enhancement in $g$ is sufficient to enter the strong coupling regime.

With any roughness at the interface of an indirect-bandgap semiconducting dot, the phase of the valley-orbit coupling can take on a random value. This random value, in double quantum dots, causes a large change in the exchange splitting. We demonstrate a simple analytical method to calculate the phase, and thus the exchange splitting and singlet-triplet qubit frequency, for an arbitrary interface. We then show that, with lateral control of the position of a quantum dot using a gate voltage, the valley-orbit phase can be controlled over a wide range, so that variations in the exchange splitting can be controlled for individual devices. Finally, we suggest experiments to measure the valley phase and the concomitant gate voltage control., Comment: 9 pages, 5 figures

We study relaxation of a moving spin qubit caused by phonon noise. As we vary the speed of the qubit, we observe several interesting features in spin relaxation and the associated phonon emission, induced by Doppler effect. In particular, in the supersonic regime, the phonons emitted by the relaxing qubit is concentrated along certain directions, similar to the shock waves produced in classical Cherenkov effect. As the speed of the moving qubit increases from the subsonic regime to the supersonic regime, the qubit experiences a peak in the spin relaxation rate near the speed of sound, which we term a spin relaxation boom in analogy to the classical sonic boom. We also find that the moving spin qubit may have a lower relaxation rate than a static qubit, which hints at the possibility of coherence-preserving transportation for a spin qubit. While the physics we have studied here has strong classical analogies, we do find that quantum confinement for the spin qubit plays an important role in all the phenomena we observe. Specifically, it produces a correction on the Cherenkov angle, and removes the divergence in relaxation rate at the sonic barrier. It is our hope that our results would encourage further research into approaches for transferring and preserving quantum information in spin qubit architectures., Comment: 8 pages, 5 figures

We study the relaxation of an electron spin qubit in a Si quantum dot due to electrical noise. In particular, we clarify how the presence of conduction-band valleys influences spin relaxation. In single-valley semiconductor quantum dots, spin relaxation is through the mixing of spin and envelope orbital states via spin-orbit interaction. In Si, the relaxation could also be through the mixing of spin and valley states. We find that the additional spin relaxation channel, via spin-valley mixing and electrical noise, is indeed important for an electron spin in a Si quantum dot. By considering both spin-valley and intravalley spin-orbit mixings and Johnson noise in a Si device, we find that the spin relaxation rate peaks at the hot spot, where the Zeeman splitting matches the valley splitting. Furthermore, because of a weaker field dependence, the spin relaxation rate due to Johnson noise could dominate over phonon noise at low magnetic fields, which fits well with recent experiments., Comment: 11 pages, 7 figures

We study decoherence of a field-driven qubit in the presence of environmental noises. For a general qubit, we find that driving, whether on-resonance or off-resonance, alters the qubit decoherence rates (including dissipation and pure dephasing), allowing both blue and red sideband contributions from the reservoir. Depending on the noise spectral density, driving field detuning and driving field phase shift, the qubit decoherence rates could be either accelerated or reduced. We apply our general theory to the system of an electron spin qubit that is confined in a quantum dot and driven by an in-plane electric field. We analyze how spin relaxation induced by the electrical noise due to electron-phonon interaction varies as a function of driving frequency, driving magnitude, driving field phase shift and spin-orbit coupling strengths., Comment: 12 pages, 7 figures

We study decoherence of an electron spin qubit in a quantum dot due to charge noise. We find that at the lowest order, the pure dephasing channel is suppressed for both $1/f$ charge noise and Johnson noise, so that charge noise leads to a pure relaxation channel of decoherence. Because of the weaker magnetic field dependence, the spin relaxation rate due to charge noise could dominate over phonon noise at low magnetic fields in a gate-defined GaAs or Si quantum dot or a InAs self-assembled quantum dot. Furthermore, in a large InAs self-assembled quantum dot, the spin relaxation rate due to phonon noise could be suppressed in high magnetic field, and the spin relaxation due to charge noise could dominate in both low and high magnetic field. Numerically, in a 1 Tesla field, the spin relaxation time due to typical charge noise is about $100$ s in Si, $0.1$ s in GaAs for a gate-defined quantum dot with a $1$ meV confinement, and $10$ $\mu$s in InAs self-assembled quantum dot with a $4$ meV confinement., Comment: 12 pages, 8 figures

Long-range quantum communication for spin qubits is an important open problem. Here we study decoherence of an electron spin qubit that is being transported in a moving quantum dot. We focus on spin decoherence due to spin-orbit interaction and a random electric potential. We find that at the lowest order, the motion induces longitudinal spin relaxation, with a rate linear in the dot velocity. Our calculated spin relaxation time ranges from sub $\mu$s in GaAs to above ms in Si, making this relaxation a significant decoherence channel. Our results also give clear indications on how to reduce the decoherence effect of electron motion., Comment: 7 pages and 3 figures

The dynamics of a qubit under the decoherence of a two level fluctuator (TLF) in addition to its coupling to a bosonic bath is investigated theoretically. Two different methods are applied and compared for this problem. One is a perturbation method based on a unitary transformation. With the merit of our unitary transformation, non-adiabatic effect can be taken into account efficiently. And the other one is the numerically exact method, namely the quasi-adiabatic path-integral (QUAPI) propagator technique. We find that the analytical method works well for a wide parameter range and show good agreement with QUAPI. On the other hand, The enhancement and the reduction of quantum decoherence of the qubit is checked with varying bath temperature $T$ and TLF-bath coupling., Comment: 5 figures

The dynamics of a qubit in a structured environment is investigated theoretically. One point of view of the model is the spin-boson model with a Lorentz shaped spectral density. An alternative view is a qubit coupled to harmonic oscillator (HO), which in turn coupled to a Ohmic environment. Two different methods are applied and compared for this problem. One is a perturbation method based on a unitary transformation. Since the transformed hamiltonian is of rotating wave approximation (RWA) form, we call it the transformed rotating wave approximation (TRWA) method. And the other one is the numerically exact method of the quasi-adiabatic propagator path-integral (QUAPI) method. TRWA method can be applied from the first point of view. And the QUAPI method can applied from both points of views. We find that from the 1st point of view QUAPI only works well for large $\Gamma$. Since the memory time is too long for the practical evaluation of QUAPI when $\Gamma$ is small. We call this treatment as QUAPI1. And from the 2nd point of view, QUAPI works well for small $\Gamma$, since the non-adiabatic effect become more important as $\Gamma$ increases, one need smaller time-step and more steps to obtain accurate result which also quickly runs out the computational resources. This treatment is called QUAPI2. We find that the TRWA method works well for the whole parameter range of $\Gamma$ and show good agreement with QUAPI1 and QUAPI2. On the other hand, we find that the decoherence of the qubit can be reduced with increasing coupling between HO and bath. This result may be relevant to the design of quantum computer., Comment: 5 figures

The dynamics of a qubit in two different environments are investigated theoretically. The first environment is a two level system coupled to a bosonic bath. And the second one is a damped harmonic oscillator. Based on a unitary transformation, we find that the decoherence of the qubit can be reduced with increasing temperature $T$ in the first case, which agree with the results in [Phys. Rev. Lett. 100, 120401], whereas, it can not be reduced with $T$ in the second case. In both cases, the qubit dynamics are changed substantially as the coupling increases or finite detuning appears., Comment: 10 pages, 5 figures

A new analytical approach, beyond rotating wave approximation, based on unitary transformations and the non-Markovian master equation for the density operator, is applied to treat the biased spin boson model with a Lorentzian structured bath for arbitrary detunings at zero temperature. Compared to zero bias, we find that the dynamics demonstrates two more damping oscillation frequencies and one additional relaxation frequency for non-zero bias, where one of the damping oscillation frequencies is a new effect. Analytical expressions for the non-Markovian dynamics and the corresponding spectrum, the localized-delocalized transition point, the coherent-incoherent transition point, the analytical ground energy, the renormalized tunneling factor and the susceptibility are determined. The sum rule and the Shiba relation are checked in the coherent regime., Comment: 10 figures, revised version, has been submitted to JPCM on Oct. 15

The dynamics of an unbiased spin-boson model with Lorentzian spectral density is investigated theoretically in terms of the perturbation theory based on a unitary transformation. The non-equilibrium correlation function $P(t)$ and susceptibility $\chi^{\prime\prime}(\omega)$ are calculated for both the off-resonance case $\Delta\lesssim 0.5\Omega$ and the on-resonance case $\Delta\sim \Omega$. The approach is checked by the Shiba's relation and the sum rule. Besides, the coherent-incoherent transition point $\alpha_c$ can be determined, which has not been demonstrated for the structured bath by previous authors up to our knowledge., Comment: 25 pages, 11 figures