Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function C（ω） of the spin operator σz for the biased sub-Ohmic spin-boson model. The small-ω behavior C...Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function C（ω） of the spin operator σz for the biased sub-Ohmic spin-boson model. The small-ω behavior C（ω） ∝ ω～s is found to be universal and independent of the bias ε and the coupling strength α（except at the quantum critical point α = αc and ε = 0）. Our NRG data also show C（ω） ∝ χ～2ω～s for a wide range of parameters, including the biased strong coupling regime（ε = 0 and α 〉 αc）, supporting the general validity of the Shiba relation. Close to the quantum critical point αc,the dependence of C（ω） on α and ε is understood in terms of the competition between ε and the crossover energy scale ω0^＊of the unbiased case. C（ω） is stable with respect to ε for ε《ε^＊. For ε 》ε^＊, it is suppressed by ε in the low frequency regime. We establish that ε^＊∝（ω0^＊）^1/θ holds for all sub-Ohmic regime 0≤s 〈 1, with θ = 2/（3s） for 0 〈 s≤1/2 and θ = 2/（1 ＋ s） for 1/2 〈 s 〈 1. The variation of C（ω） with α and ε is summarized into a crossover phase diagram on the α–ε plane.展开更多
The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method.We focus on the dynamical auto-correlation functions CO（ω）, with the operator taken as σx, σz, and ...The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method.We focus on the dynamical auto-correlation functions CO（ω）, with the operator taken as σx, σz, and X, respectively. In the weak-coupling regime α 〈 αc, these functions show power law ω-dependence in the small frequency limit, with the powers 1 ＋ 2s, 1 ＋ 2s, and s, respectively. At the critical point α = αc of the boson-unstable quantum phase transition, the critical exponents yO of these correlation functions are obtained as yσx= yσz= 1-2s and yX=-s, respectively. Here s is the bath index and X is the boson displacement operator. Close to the spin flip point, the high frequency peak of Cσx（ω） is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.展开更多
In this paper,we explore how to estimate the phase damping parameter γ and the tunneling amplitude parameter ？from a spin-boson dephasing quantum model by periodical projective measurements.The preparation of initia...In this paper,we explore how to estimate the phase damping parameter γ and the tunneling amplitude parameter ？from a spin-boson dephasing quantum model by periodical projective measurements.The preparation of initial states is accomplished by performing the period measurements in our scheme.The parameter γ can be always estimated when projective measurement bases are chosen as θ = π/2 and φ = 0.Based on the estimated value of γ and the interval information of ？,we can select another measurement bases（θ = π/4 and φ = π/2） to obtain the estimated value of ？.A coherent control is indispensable to estimate ？ if γ is in the interval of ？;whereas the control is not necessary if γ is out of the known interval of ？.We establish the relation between the optimal period time and the parameter γ or ？ in terms of Fisher information.Although the optimal measurement period cannot be selected beforehand,the aforementioned relation can be utilized to adjust the measurement period to approach the optimal one.展开更多
We present mathematical analyses of the evolution of solutions of the self-consistent equation derived from variational calculations based on the displaced-oscillator-state and the displaced-squeezed-state in spin-bos...We present mathematical analyses of the evolution of solutions of the self-consistent equation derived from variational calculations based on the displaced-oscillator-state and the displaced-squeezed-state in spin-boson model at a zero temperature and a finite temperature. It is shown that, for a given spectral function defined as J（w） = π∑k Ck^2 = π/2αw^8w^1-s, there exists a universal sc for both kinds of variational schemes, the localized transition happens only for 2 s ≤ sc, moreover, the localized transition is discontinuous for s 〈 sc while a continuous transition always occurs when s = sc. At T = 0, we have sc = 1, while for T ≠ 0, sc = 2 which indicates that the localized transition in super-Ohmic case still exists, manifesting that the result is in discrepancy with the existing result.展开更多
We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σ_(z) at finite temperature for the sub-ohmic spin-boson mo...We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σ_(z) at finite temperature for the sub-ohmic spin-boson model.A peak is observed at the frequency ω_(T)~T in the curve of C(ω).The curve merges with the zero-temperature C(ω) in ω>>ω_(T) and deviates significantly from the zero-temperature curve in ω<<ω_(T).展开更多
We investigate the dynamics of a system coupled to an environment by averaged semiquantum method.The theory origins from the time-dependent variational principle(TDVP)formulation and contains nondiagonal matrixelement...We investigate the dynamics of a system coupled to an environment by averaged semiquantum method.The theory origins from the time-dependent variational principle(TDVP)formulation and contains nondiagonal matrixelements.So it can be applied to study dissipation,measurement,and decoherence problems in the model(H=h_S+h_E+h_I).In the calculation,the influence of the environment govern by differential dynamical equation is incorporatedthrough a mean Geld.We have performed averaged semiquantum method for a spin-boson model,which reproduce theresults from stochastic Schrodinger equation method and Hierarchical approach quite accurately.The problems,dynamicsin nonequilibrium environments,have also been studied by our method.展开更多
We study the evolution properties of spin-boson systems by a systematic numerical iteration approach,which performs well in the whole coupling regime.This approach evaluates a set of coefficients in the formal express...We study the evolution properties of spin-boson systems by a systematic numerical iteration approach,which performs well in the whole coupling regime.This approach evaluates a set of coefficients in the formal expression of the time-dependent Schr?dinger equation by expanding the initial state in Fock space.This set of coefficients is unique for the spin-boson Hamiltonian studied,allowing one to calculate the time evolution from different initial states.To complement our numerical calculations,we apply the method to the Buck–Sukumar model.We find that when the ground-state energy of the model is unbounded and no ground state exists in a certain parameter space,the time evolution of the physical quantities is naturally unstable.展开更多
Based on the rapid experimental developments of circuit QED,we propose a feasible scheme to simulate the spin-boson model with superconducting circuits,which can be used to detect quantum Kosterlitz-Thouless(KT) phase...Based on the rapid experimental developments of circuit QED,we propose a feasible scheme to simulate the spin-boson model with superconducting circuits,which can be used to detect quantum Kosterlitz-Thouless(KT) phase transition.We design the spinboson model by using a superconducting phase qubit coupled to a semi-infinite transmission line,which is regarded as a bosonic reservoir with a continuum spectrum.By tuning the bias current or the coupling capacitance,the quantum KT transition can be directly detected through tomography measurement on the states of the phase qubit.We also estimate the experimental parameters using the numerical renormalization group method.展开更多
We explore the spin-boson model in a special case, i.e., with zero local field. In contrast to previous studies, we find no possibility for quantum phase transition(QPT) happening between the localized and delocalized...We explore the spin-boson model in a special case, i.e., with zero local field. In contrast to previous studies, we find no possibility for quantum phase transition(QPT) happening between the localized and delocalized phases, and the behavior of the model can be fully characterized by the even or odd parity as well as the parity breaking, instead of the QPT, owned by the ground state of the system. The parity breaking mentioned in our case is completely diferent from the spontaneously broken symmetry relevant to the conventionally defined QPT in previous studies. Our analytical treatment about the eigensolution of the ground state of the model presents for the first time a rigorous proof of nodegeneracy for the ground state of the model, which is independent of the bath type, the degrees of freedom of the bath and the calculation precision. We argue that the QPT mentioned previously appears due to incorrect employment of the ground state of the model and/or unreasonable treatment of the infrared divergence existing in the spectral functions for Ohmic and sub-Ohmic dissipations.展开更多
In many previous temperature estimation schemes,the temperature of a sample is directly read out from the final steady state of a quantum probe,which is coupled to the sample.However,in these studies,information of co...In many previous temperature estimation schemes,the temperature of a sample is directly read out from the final steady state of a quantum probe,which is coupled to the sample.However,in these studies,information of correlations between system(the probe)and reservoir(the sample)is usually eliminated,leading the steady state of the probe is a canonical equilibrium state with respect solely to system's Hamiltonian.To explore the influence of system-reservoir correlations on the estimation precision,we investigate the equilibration dynamics of a spin interacting with a finite temperature bosonic reservoir.By incorporating an intermediate harmonic oscillator or a collective coordinate into the spin,the system-reservoir correlations can be correspondingly encoded in a Gibbs state of an effective Hamilton,which is size consistent with the original bare spin.Extracting information of temperature from this corrected steady state,we find the effect of the system-reservoir correlations on the estimation precision is highly sensitive to the details of the spectral density function of the measured reservoir.展开更多
基金supported by the National Basic Research Program of China(Grant No.2012CB921704)the National Natural Science Foundation of China(Grant No.11374362)+1 种基金the Fundamental Research Funds for the Central Universities,Chinathe Research Funds of Renmin University of China(Grant No.15XNLQ03)
文摘Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function C（ω） of the spin operator σz for the biased sub-Ohmic spin-boson model. The small-ω behavior C（ω） ∝ ω～s is found to be universal and independent of the bias ε and the coupling strength α（except at the quantum critical point α = αc and ε = 0）. Our NRG data also show C（ω） ∝ χ～2ω～s for a wide range of parameters, including the biased strong coupling regime（ε = 0 and α 〉 αc）, supporting the general validity of the Shiba relation. Close to the quantum critical point αc,the dependence of C（ω） on α and ε is understood in terms of the competition between ε and the crossover energy scale ω0^＊of the unbiased case. C（ω） is stable with respect to ε for ε《ε^＊. For ε 》ε^＊, it is suppressed by ε in the low frequency regime. We establish that ε^＊∝（ω0^＊）^1/θ holds for all sub-Ohmic regime 0≤s 〈 1, with θ = 2/（3s） for 0 〈 s≤1/2 and θ = 2/（1 ＋ s） for 1/2 〈 s 〈 1. The variation of C（ω） with α and ε is summarized into a crossover phase diagram on the α–ε plane.
基金supported by the National Key Basic Research Program of China(Grant No.2012CB921704)the National Natural Science Foundation of China(Grant No.11374362)+1 种基金the Fundamental Research Funds for the Central Universities,Chinathe Research Funds of Renmin University of China(Grant No.15XNLQ03)
文摘The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method.We focus on the dynamical auto-correlation functions CO（ω）, with the operator taken as σx, σz, and X, respectively. In the weak-coupling regime α 〈 αc, these functions show power law ω-dependence in the small frequency limit, with the powers 1 ＋ 2s, 1 ＋ 2s, and s, respectively. At the critical point α = αc of the boson-unstable quantum phase transition, the critical exponents yO of these correlation functions are obtained as yσx= yσz= 1-2s and yX=-s, respectively. Here s is the bath index and X is the boson displacement operator. Close to the spin flip point, the high frequency peak of Cσx（ω） is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61673389,61273202,and 61134008)
文摘In this paper,we explore how to estimate the phase damping parameter γ and the tunneling amplitude parameter ？from a spin-boson dephasing quantum model by periodical projective measurements.The preparation of initial states is accomplished by performing the period measurements in our scheme.The parameter γ can be always estimated when projective measurement bases are chosen as θ = π/2 and φ = 0.Based on the estimated value of γ and the interval information of ？,we can select another measurement bases（θ = π/4 and φ = π/2） to obtain the estimated value of ？.A coherent control is indispensable to estimate ？ if γ is in the interval of ？;whereas the control is not necessary if γ is out of the known interval of ？.We establish the relation between the optimal period time and the parameter γ or ？ in terms of Fisher information.Although the optimal measurement period cannot be selected beforehand,the aforementioned relation can be utilized to adjust the measurement period to approach the optimal one.
基金supported by the National Natural Science Foundation of China (Grant No 10575045)
文摘We present mathematical analyses of the evolution of solutions of the self-consistent equation derived from variational calculations based on the displaced-oscillator-state and the displaced-squeezed-state in spin-boson model at a zero temperature and a finite temperature. It is shown that, for a given spectral function defined as J（w） = π∑k Ck^2 = π/2αw^8w^1-s, there exists a universal sc for both kinds of variational schemes, the localized transition happens only for 2 s ≤ sc, moreover, the localized transition is discontinuous for s 〈 sc while a continuous transition always occurs when s = sc. At T = 0, we have sc = 1, while for T ≠ 0, sc = 2 which indicates that the localized transition in super-Ohmic case still exists, manifesting that the result is in discrepancy with the existing result.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11374362 and 11974420)the Fundamental Research Funds for the Central Universities,Chinathe Research Funds of Renmin University of China(Grant No.15XNLQ03)。
文摘We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σ_(z) at finite temperature for the sub-ohmic spin-boson model.A peak is observed at the frequency ω_(T)~T in the curve of C(ω).The curve merges with the zero-temperature C(ω) in ω>>ω_(T) and deviates significantly from the zero-temperature curve in ω<<ω_(T).
基金Supported by the National Natural Science Foundation under Grant Nos.1037504 and 10875087
文摘We investigate the dynamics of a system coupled to an environment by averaged semiquantum method.The theory origins from the time-dependent variational principle(TDVP)formulation and contains nondiagonal matrixelements.So it can be applied to study dissipation,measurement,and decoherence problems in the model(H=h_S+h_E+h_I).In the calculation,the influence of the environment govern by differential dynamical equation is incorporatedthrough a mean Geld.We have performed averaged semiquantum method for a spin-boson model,which reproduce theresults from stochastic Schrodinger equation method and Hierarchical approach quite accurately.The problems,dynamicsin nonequilibrium environments,have also been studied by our method.
基金supported by the National Natural Science Foundation of China under Grant Nos.11835011 and 11774316。
文摘We study the evolution properties of spin-boson systems by a systematic numerical iteration approach,which performs well in the whole coupling regime.This approach evaluates a set of coefficients in the formal expression of the time-dependent Schr?dinger equation by expanding the initial state in Fock space.This set of coefficients is unique for the spin-boson Hamiltonian studied,allowing one to calculate the time evolution from different initial states.To complement our numerical calculations,we apply the method to the Buck–Sukumar model.We find that when the ground-state energy of the model is unbounded and no ground state exists in a certain parameter space,the time evolution of the physical quantities is naturally unstable.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11004065,11104057 and 11125417)the Natural Science Foundation of Guangdong Province (Grant No.10451063101006312)+1 种基金the State Key Program for Basic Research of China(Grant No. 2011CB922104)the GRF and CRF of the RGC of Hong Kong
文摘Based on the rapid experimental developments of circuit QED,we propose a feasible scheme to simulate the spin-boson model with superconducting circuits,which can be used to detect quantum Kosterlitz-Thouless(KT) phase transition.We design the spinboson model by using a superconducting phase qubit coupled to a semi-infinite transmission line,which is regarded as a bosonic reservoir with a continuum spectrum.By tuning the bias current or the coupling capacitance,the quantum KT transition can be directly detected through tomography measurement on the states of the phase qubit.We also estimate the experimental parameters using the numerical renormalization group method.
基金Supported by National Fundamental Research Program of China under Grant No.2012CB922102National Natural Science Foundation of China under Grant Nos.10974225 and 11004226State Key Laboratory Funding of WIPM
文摘We explore the spin-boson model in a special case, i.e., with zero local field. In contrast to previous studies, we find no possibility for quantum phase transition(QPT) happening between the localized and delocalized phases, and the behavior of the model can be fully characterized by the even or odd parity as well as the parity breaking, instead of the QPT, owned by the ground state of the system. The parity breaking mentioned in our case is completely diferent from the spontaneously broken symmetry relevant to the conventionally defined QPT in previous studies. Our analytical treatment about the eigensolution of the ground state of the model presents for the first time a rigorous proof of nodegeneracy for the ground state of the model, which is independent of the bath type, the degrees of freedom of the bath and the calculation precision. We argue that the QPT mentioned previously appears due to incorrect employment of the ground state of the model and/or unreasonable treatment of the infrared divergence existing in the spectral functions for Ohmic and sub-Ohmic dissipations.
基金Project supported by the National Natural Science Foundation of China(Grants Nos.11704025,11674139,and 11834005).
文摘In many previous temperature estimation schemes,the temperature of a sample is directly read out from the final steady state of a quantum probe,which is coupled to the sample.However,in these studies,information of correlations between system(the probe)and reservoir(the sample)is usually eliminated,leading the steady state of the probe is a canonical equilibrium state with respect solely to system's Hamiltonian.To explore the influence of system-reservoir correlations on the estimation precision,we investigate the equilibration dynamics of a spin interacting with a finite temperature bosonic reservoir.By incorporating an intermediate harmonic oscillator or a collective coordinate into the spin,the system-reservoir correlations can be correspondingly encoded in a Gibbs state of an effective Hamilton,which is size consistent with the original bare spin.Extracting information of temperature from this corrected steady state,we find the effect of the system-reservoir correlations on the estimation precision is highly sensitive to the details of the spectral density function of the measured reservoir.
