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arXiv:quant-ph/0007007 · Physical Review A

Stochastic limit approximation for rapidly decaying systems

Gen Kimura, Kazuya Yuasa, Kentaro Imafuku

Published Jan 10, 2001• 7 citations
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Abstract

The stochastic limit approximation method for ``rapid'' decay is presented, where the damping rate $\ensuremath{\gamma}$ is comparable to the system frequency $\ensuremath{\Omega},$ i.e., $\ensuremath{\gamma}\ensuremath{\sim}\ensuremath{\Omega},$ whereas the usual stochastic limit approximation is applied only to the weak damping situation $\ensuremath{\gamma}\ensuremath{\ll}\ensuremath{\Omega}.$ The key formulas for rapid decay are very similar to those for weak damping, but the dynamics are quite different. From a microscopic Hamiltonian, the spin-boson model, a Bloch equation containing two independent time scales is derived. This is a useful method to extract the minimal dissipative dynamics at high temperature ${k}_{B}T\ensuremath{\gg}\ensuremath{\Elzxh}\ensuremath{\Omega}$ and the master equations obtained are of the Lindblad form unlike that of Caldeira and Leggett. The validity of the method is confirmed by comparing the master equation derived through this method with the exact one.

DOI
10.1103/physreva.63.022103

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The stochastic limit approximation method for ``rapid'' decay is presented, where the damping rate \ensuremathγ\ensuremath{\gamma}\ensuremathγ is comparable to the system frequency \ensuremathΩ,\ensuremath{\Omega},\ensuremathΩ, i.e., \ensuremathγ\ensuremath∼\ensuremathΩ,\ensuremath{\gamma}\ensuremath{\sim}\ensuremath{\Omega},\ensuremathγ\ensuremath∼\ensuremathΩ, whereas the usual stochastic limit approximation is applied only to the weak damping situation \ensuremathγ\ensuremath≪\ensuremathΩ.\ensuremath{\gamma}\ensuremath{\ll}\ensuremath{\Omega}.\ensuremathγ\ensuremath≪\ensuremathΩ. The key formulas for rapid decay are very similar to those for weak damping, but the dynamics are quite different. From a microscopic Hamiltonian, the spin-boson model, a Bloch equation containing two independent time scales is derived. This is a useful method to extract the minimal dissipative dynamics at high temperature kBT\ensuremath≫\ensuremath\Elzxh\ensuremathΩ{k}_{B}T\ensuremath{\gg}\ensuremath{\Elzxh}\ensuremath{\Omega}kB​T\ensuremath≫\ensuremath\Elzxh\ensuremathΩ and the master equations obtained are of the Lindblad form unlike that of Caldeira and Leggett. The validity of the method is confirmed by comparing the master equation derived through this method with the exact one.
Master equation of the Lindblad form based on amicroscopic Hamiltonian through stochastic limit approximation -For rapidly decaying systems -Accardi et al. の確率極限近似 (stochastic limit approximation) [7,8][7,8][7,8] は, 全体系から散逸ダイナ ミクスを引き出す有力な手法である.それが扱うのは, 次のような全体系 Hamiltonian である: Htot=HS+λV+HBH_{\mathrm{t}\mathrm{o}\mathrm{t}}=H_{S}+\lambda V+H_{B}Htot​=HS​+λV+HB​ .(1) 例えぼ, 電磁場 HBH_{B}HB​ 中の原子 HsHsHs を考えよ.量子光学においてしぼしば興味のある状況である. そ の場合, 電磁相互作用 λV\lambda VλV は十分弱く (λ≪1)(\lambda\ll 1)(λ≪1) , 摂動計算は十分に良い近似を与えるであろう. 確 率極限近似は, 同時に時間の粗視化 t↦τ=λ2tt\mapsto\tau=\lambda^{2}tt↦τ=λ2t を行い, 弱結合極限 (weak coupling limit) $\lambdaarrow 0$ ( τ\tauτ 固定) によって摂動の最低次の寄与を抽出することで, 最小限ではあるが十分に興味深い散逸ダ イナミクスを導く方法である [9-11].しかしながら, Hamiltonian (1) {?}こ確率極限近似を適用して導かれるダイナミクスは, 少々状況 が限定されている.一般に, 緩和定数 γ\gammaγ に対する相互作用の最低次の寄与は λ2\lambda^{2}λ2 に比例し, 注目系HsHsHs の特徴的振動数 Ω\OmegaΩ に比べて γ<<Ω\gamma<<\Omegaγ<<Ω .すなわち, 弱い散逸 ( WD\mathrm{W}\mathrm{D}WD :Weak Damping) が導かれる のである.量子光学はこの状況にあたる.これに比べて緩和が速い γ∼Ω\gamma\sim\Omegaγ∼Ω という状況 ( RD\mathrm{R}\mathrm{D}RD :Rapid Decay) を導こうと思ったら, 別の扱いが必要なのである.また, RD\mathrm{R}\mathrm{D}RD の取り扱いは, WD\mathrm{W}\mathrm{D}WD に比べ て少々注意を要することが指摘されている [2,12][2,12][2,12] .WD\mathrm{W}\mathrm{D}WD においては通常 Lindblad 型マスター方程 式 [13] が導かれている [2,6][2,6][2,6] -方で, RD\mathrm{R}\mathrm{D}RD の状況において導出されるマスタ -方程式は, しぼしぼ Lindblad 型でないのである.そのため, RD\mathrm{R}\mathrm{D}RD のマスター方程式はしぼしぼ, 確率の正値性が保証さ れないという困難を伴うことになる [2,12][2,12][2,12] .1
Abstract Realistic models of quantum systems must include dissipative interactions with a thermal environment. For weakly-damped systems, while the Lindblad-form Markovian master equation is invaluable for this task, it applies only when the frequencies of any subset of the system’s transitions are degenerate, or their differences are much greater than the transitions’ linewidths. Outside of these regimes the only available efficient description has been the Bloch–Redfield master equation, the efficacy of which has long been controversial due to its failure to guarantee the positivity of the density matrix. The ability to efficiently simulate weakly-damped systems across all regimes is becoming increasingly important, especially in quantum technologies. Here we solve this long-standing problem by deriving a Lindblad-form master equation for weakly-damped systems that is accurate for all regimes. We further show that when this master equation breaks down, so do all time-independent Markovian equations, including the B-R equation. We thus obtain a replacement for the B-R equation for thermal damping that is no less accurate, simpler in structure, completely positive, allows simulation by efficient quantum trajectory methods, and unifies the previous Lindblad master equations. We also show via exact simulations that the new master equation can describe systems in which slowly-varying transition frequencies cross each other during the evolution. System identification tools, developed in systems engineering, play an important role in our analysis. We expect these tools to prove useful in other areas of physics involving complex systems.
The dynamics of the spin-boson Hamiltonian is considered in the stochastic approximation. The Hamiltonian describes a two-level system coupled to an environment, and is widely used in physics, chemistry, and the theory of quantum measurement. We demonstrate that the method of stochastic approximation, a general method for consideration of dynamics of an arbitrary system interacting with environment, is powerful enough to reproduce qualitatively the striking results by Leggett et al. [Rev. Mod. Phys. 59, 1 (1987)] found earlier for this model system. The results include an exact expression of the dynamics in terms of the spectral density, and show the appearance of two interesting regimes for the system, i.e., pure oscillating and pure damping ones. Correlators describing the environment are also computed.
We develop a hierarchical functional derivative (HFD) method to investigate the reduced dynamics of a spin-boson model within the framework of a stochastic decoupling description. Keeping only the lowest-order truncation of the hierarchical functional derivatives, one can recover the second-order Nakajima-Zwanzig quantum master equation. Taking into account the higher-order corrections, our method can be implemented as a highly efficient numerical scheme for a general spin-boson model beyond the usual Markovian, rotating-wave, and weak-coupling approximations.