Quantum-mechanical derivation of the Bloch equations: Beyond the weak-coupling limit

Abstract

Two nondegenerate quantum levels coupled off-diagonally and linearly to a bath of quantummechanical harmonic oscillators are considered. In the weak-coupling limit one finds that the equations of motion for the reduced density-matrix elements separate naturally into two uncoupled pairs of linear equations for the diagonal and off-diagonal elements, which are known as the Bloch equations. The equations for the populations form the simplest twocomponent master equation, and the rate constant for the relaxation of nonequilibrium population distributions is liT), defined as the sum of the "up" and "down" rate constants in the master equation. Detailed balance is satisfied for this master equation in that the ratio of these rate constants is equal to the ratio of the equilibrium popUlations. The relaxation rate constant for the off-diagonal density-matrix elements is known as l/T 2 . One finds that this satisfies the well-known relation l/T2 = 1/2TI' In this paper the weak-coupling limit is transcended by deriving the Bloch equations to fourth order in the coupling. The equations have the same form as in the weak-coupling limit, but the rate constants are calculated to fourth order. For the population-relaxation rate constants this results in an extension to fourth order of Fermi's golden rule. We find that these higher-order rate constants do indeed satisfy detailed balance. Comparing the dephasing and population-relaxation rate constants, we find that in fourth order lIT 2 # lI2T!.

Quantum thermodynamic cooling cycle

“…withC,S being real andC ≥ 0 [9,15,17]. With Eq (A3), we split each term of Eq (A1) into two parts related toC andS.…”

“…The relation of this maximum cooling rate to the energy mode density has been established, and the basic result has been shown to respect the fundamental bound that was recently established for maximum cooling rate from the Second and Third Laws. To obtain the evolution equation for the 3-level system, we invoke standard tools for quantum open systems: the Redfield approach [14,15] and the secular approximation [11,16].…”

Non-Markovian dynamics and phonon decoherence of a double quantum dot charge qubit

“…Thorwart et al [14] investigated the PCPB case and they obtained the quality factor instead of the decoherence time. By using a set of parameters of the DQD charge qubit and the PCPB they obtained the quality factor of the qubit as Q pz = 336, which corresponds to decoherence time τ pz 2 = Q pz π/ω ′ pz ≈ 115.9 ps, where ω ′ = ω 0 + ∆ω, and ∆ω is the bath-induced shift [22] in the natural frequency ω 0 = 2T c . From Fig.1 of Ref.…”

“…In this method, the relaxation and dephasing times can be evaluated from the spin-bosonic model with Bloch equations [17,18]. For our model, they are [22] …”

A continued fraction resummation form of bath relaxation effect in the spin-boson model

“…7 The non-Markovian relaxation of the surrounding bath can significantly slow down the transfer process compared to the second-order prediction. [8][9][10][11][12][13][14][15][16][17] On the other hand, the transfer rate can be optimized at an intermediate dissipation strength in a biased two-site system, which can be further related to the energy transfer optimization in multi-site systems. [18][19][20][21][22][23][24] Within the single excitation manifold, the two-site system can be viewed as an extension of the spin-boson model, with possible variations in the boson bath and the bath spatial correlation.…”