Markovian master equations: a critical study

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For small temperatures, τ_B (FWHH) is quite large, so the Markovian approximation is expected to break down for values of T such that τ_S ≪ τ_B.

“From figure 4, one sees that for small temperatures, τ B (i.e.FWHH) is quite large, so it is expected that the Markovian approximation breaks down for values of T such that τ S τ B .”

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As temperature increases further, the influence of the correlation function becomes more important and the FWHH decreases to a limiting value; the accuracy at large temperatures stabilizes to only a little worse than at T=0.

“As the temperature increases further, the influence of this correlation function becomes more important and the FWHH decreases to a limiting value (see the plot on the right of figure 4); this convergence is reflected by the red, cyan and purple lines, which show that the accuracy at large temperatures stabilizes to only a little worse than that at T = 0, as was expected from figure 4.”

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The spreading of the correlation function is mainly caused by its height decrease, and in the limit T → 0, C(s,T) → 0.

“Note that the spreading of the correlation function is mainly caused by its 'height' decrease, that is, in the limit T → 0, C(s, T ) → 0.”

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At zero temperature the results are in closest agreement; as temperature increases to T=0.1, the correlation function broadens, leading to a small degradation in modelling precision.

“At zero temperature (blue line), the results are in closest agreement; however, as the temperature increases to T = 0.1, the correlation function broadens, which leads to a degradation (albeit small) in the modelling precision.”

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C(s=0,T=0.05)=3.27391×10^-7 and C0(s=0)=0.018, so C(s,T=0.05) will not have a large effect on the dynamics.

“We may get a very informal idea of how both contributions matter by looking at their maximum values at s = 0, for example C(s = 0, T = 0.05) = 3.27391 × 10 -7 and C 0 (s = 0) = 0.018, and so it is clear that C(s, T = 0.05) will not have a large effect on the dynamics.”

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The time after which revivals arise increases with the number of oscillators in the bath and roughly scales as τ_R ∝ M.

“Of course, the time after which these revivals arise increases with the number of oscillators in the bath, and roughly speaking it scales as τ R ∝ M.”

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The range where neither Markovian master equation obtains high precision becomes smaller as the coupling with the bath decreases, so generally both master equations cover a good range of values of β.

“However, note that this range becomes smaller as the coupling with the bath decreases, and so generally both master equations cover a good range of values of β.”

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The behaviour at very early times is mainly related to the choice of the initial state and reflects how it adjusts to the state of the bath under Markovian evolution, with different tendencies depending on the initial state.

“The behaviour at very early times is mainly related to the choice of the initial state of the system, and it reflects how it adjusts to the state of the bath under the Markovian evolution (for more on this topic, see for example [33]); different tendencies have been found depending on the choice of the initial state.”

• cites [33] sourceNot checked yet

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The behaviour with temperature is visible at longer times, since τ_B ∼ FHWW increases with T, in agreement with the conclusions drawn from the correlation functions.

“However, the behaviour with temperature is visible at longer times (since τ B ∼ FHWW increases with T ), which is in agreement with the conclusions drawn from the correlation functions (see the small subplot).”

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The distance between the simulated whole state ρ(t) and the ansatz ρ_S(t) ⊗ ρ_th is monotonically increasing in time, strictly in the limit of an environment with infinite degrees of freedom, and its slope decreases with coupling strength.

“It is clear that this distance is monotonically increasing in time (strictly, in the limit of an environment with infinite degrees of freedom), and the slope decreases with coupling strength.”

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Equations (58) and (59) preserve both complete positivity and Gaussianity.

“By virtue of the derivation, equations ( 58) and (59) preserve both complete positivity and Gaussianity (because they arise from a linear interaction with the environment).”

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The last quantity in equation (B.5) is a sum of individual terms for each bath, which under the standard procedure of section 1.1 lead to local dissipators D1 and D2 and shifts of (58).

“The last quantity in the above equation is just a sum of the individual terms for each bath, which lead, under the standard procedure of section 1.1, to the (interaction picture) local dissipators D 1 and D 2 and shifts of (58).”

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Equation (73) gives good results due to the small Rabi frequency; however, in comparison with (75) the accuracy quickly drops off as one moves away from ω_L - Ω = 0.

“Equation (73) gives good results due to the small Rabi frequency; however, note that in comparison with (75) the accuracy quickly drops off as we move away from ω L -= 0.”

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In some cases, while the steady state of (58) is not strictly thermal, the fidelity with that of (59) is more than 99.999%.

“In some cases, while the steady state of (58) is not strictly thermal, the fidelity with that of (59) is more than 99.999%.”

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At resonance ω_L=Ω, ã(t)=e^{-iΩt}(a+i r t), which is not periodic, so the desired decomposition as a sum of exponentials with time-independent coefficients does not exist.

“Solving equation (B.19) under this condition, we find ã(t) = e -i t (air t), (B.24) and so one can see that ã(t) is not a periodic function, so the desired decomposition as a sum of exponentials with time-independent coefficients does not exist.”

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A similar effect can be seen when compared with (74) for larger detunings.

“A similar effect can be seen when compared with (74) for larger detunings.”

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At later times the behaviour of both equations starts to differ, and the steady states are not the same.

“However, note that at later times the behaviour of both equations starts to differ, and the steady states are not the same.”

The quote only partly states this claim.

The operator Ã1(t)=ã(t) satisfies a differential equation with periodic terms iȧ(t)=[ã(t),H0(t)]=Ωã(t)+Ωr e^{-iωL t}.

“Note, however, that the operator Ã1 (t) = ã(t) satisfies a differential equation with periodic terms i ȧ(t) = [ ã(t), H 0 (t)] = ã(t) + r e -iω L t .”

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Under the assumption that detuning is large enough |ω_L-Ω| >> α, |ω_L-Ω| >> 2αr, the secular approximation can be made and yields master equation (B.21).

“Next we assume that the detuning is large enough |ω L -| α, |ω L -| 2 αr in order to make the secular approximation, and after some tedious, but straightforward, algebra we find the master equation in the interaction picture to be”

• AssumptionThe authors state this as an assumption: The detuning is large enough |ω_L-Ω| >> α, |ω_L-Ω| >> 2αr to make the secular approximation.

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For global features, the more elaborate equation (75) works better in both cases, although the difference from (73) is very small.

“For global features, the more elaborate equation (75) works better in both cases, although the difference from ( 73) is very small.”

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The choice of (74) is preferable to (73) when out of resonance, but gives quite poor results when close to resonance.

“As expected, the choice of (74) is preferable to the choice of (73) when out of resonance, but gives quite poor results when close to resonance.”

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The steady state of equation (58) is not thermal, although it tends to the thermal state as β → 0.

“equation (58) is not (although it tends to the thermal state as β → 0, of course).”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

In every case the covariance matrix is unaffected by the driving term, which only produces a change in the first moments.

“In addition, note that in every case the covariance matrix is unaffected by the driving term, which only produces a change in the first moments.”

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Weak coupling implies α << β, and to second order in α and β, K2 has the same form as (6).

“If we consider the second kernel, weak coupling implies α β, and so to second order in α and β this becomes K 2 (t, s) = α 2 PV SB (t)QV SB (s)P + O(α 3 , α 2 β), which has exactly the same form as (6)”

• AssumptionThe authors state this as an assumption: Weak coupling implies α << β.

Probably not a substantial claim: may not pass S3 (load-bearing).

At first order in r and coupling α, equation (73) is obtained, as expected from arguments in section 3.2.1.

“It is worth noting that at first order in r and the coupling α, we obtain equation (73).This is as expected, given the arguments in section 3.2.1.”

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The rotated operators b1 and b2 satisfy standard bosonic commutation rules and represent the decomposition of the oscillatory system in normal modes.

“The new operators satisfy the standard bosonic commutation rules [b i , b † j ] = δ i j , and so this is nothing more than the decomposition of an oscillatory system in normal modes.”

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For Ω = ω_L, the solution to equation (B.19) with initial condition ã(0)=a is ã(t)=... and is periodic, yielding decomposition Ã1(t)=A1(ωL)e^{-iωL t}+A2(Ω)e^{-iΩ t}.

“The solution to equation (B.19), with the initial condition ã(0) = a and for = ω L , is given by ã(t) = r (e -iω L t e -i t ) + a(ω L -)e -i t ω L -, (B.20) so in this case the solution is periodic and the desired decomposition”

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Equations (74) and (75) work better except in the case of resonance, where (73) gives more accurate results, as long as the Rabi frequency is small.

“Equations ( 74) and (75) work better except in the case of resonance, where (73) gives more accurate results, as long as the Rabi frequency is small.”

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The interactions individually satisfy Tr_B1[Ṽ1B1 ρth1]=Tr_B2[Ṽ2B2 ρth2]=0, so PV1B1V2B2P=PV2B2V1B1P=0.

“because our interactions individually hold Tr B1 [ Ṽ1B1 ρ th1 ] = Tr B2 [ Ṽ2B2 ρ th2 ] = 0, so PV 1B1 V 2B2 P = PV 2B2 V 1B1 P = 0”

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In the resonance case, only a commutator term arises from cross terms in the analogue of equation (B.15), so positivity is not lost.

“However, note that in this particular case, only a commutator term arises from the cross terms in the analogue of equation (B.15), so positivity is not lost.”

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For large intercoupling β, the non-secular terms oscillate with phase e±2iβt, so to neglect them one must impose β ≫ α; the resultant equation is complementary to (58), valid if α ≫ β.

“the non-secular terms oscillate with a phase e ±2iβt , so in order to neglect them we must impose β α; therefore the resultant equation is, in some sense, complementary to ( 58), valid if α β.”

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For the chosen ohmic spectral density with exponential cut-off, J(ω) increases linearly for small ω, decays exponentially for large ω, and has its maximum at ω = ω_c.

“Clearly, J (ω) increases linearly for small values of ω, decays exponentially for large ones and has its maximum at ω = ω c .”

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Equation (59) is expected to remain valid for small β, because when β → 0 it approaches (58) if the bath temperatures and spectral densities are the same.

“In the case of equation (59), it is reasonable to expect that this will remain valid for small β, because when β → 0, this equation approaches (58) if the bath temperatures and spectral densities are the same.”

Classified as probably not a substantial claim.

Gaussian states are entirely characterized by their first and second moments.

“This is because the harmonic oscillator falls into a class of quantum states known as Gaussian states, which are entirely characterized by their first and second moments.”

• cites [7] sourceNot checked yet

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Equation (15) generates a completely positive semigroup and defines a proper Markovian master equation.

“With this final remark, we conclude that equation (15) generates a completely positive semigroup [6] and so defines a proper Markovian master equation, i.e. a completely positive semigroup.”

• cites [6] Breuer H-P, 2002, The Theory of Open Quantum SystemsNot checked yet

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The weak coupling approach makes sense if the coupling is small and the environment has infinite degrees of freedom.

“In section 1.1, we pointed out that the weak coupling approach makes sense if the coupling is small and the environment has infinite degrees of freedom.”

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A Markovian (Kossakowski-Lindblad) master equation is required to be trace preserving and to satisfy complete positivity.

“Such a differential equation, the so-called Markovian (or Kossakowski-Lindblad) master equation, is required to fulfil several consistency properties, such as being trace preserving and satisfying complete positivity [2]- [6], [8]- [10].”

• cites [2]- [6], [8]- [10]Not checked yet

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For the Markovian approximation to be valid, the typical time scale for the evolution of the system due to its interaction with the bath τ_S must be large compared with the decay time τ_B of the correlation functions.

“In order to make valid the Markovian approximation, the typical time scale for the evolution of the system due to its interaction with the bath τ S must be large in comparison with the decay time τ B of the correlation functions.”

Probably not a substantial claim: may not pass S3 (load-bearing).

An exact master equation for two interacting harmonic oscillators subject to a global general environment has recently been derived.

“Recently, an exact master equation for two interacting harmonic oscillators subject to a global general environment was derived [23].”

• cites [23] Carmichael H J, 1999, Statistical Methods in Quantum Optics. I: Master Equations and Fokker-Planck Equations sourceNot checked yet

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The eigenfrequencies of the normal modes become imaginary if ω < 2β, and the system then becomes unstable, so β must be limited to keep oscillatory behaviour.

“However, note that the eigenfrequencies of the normal modes become imaginary if ω < 2β (see for example [34]) and the system then becomes unstable, so even when keeping the anti-rotating terms, we must limit β if we wish to keep the oscillatory behaviour.”

• cites [34] sourceNot checked yet

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A Gaussian characteristic function, and therefore any Gaussian state, can be completely specified by 2n^2 + n real parameters.

“Thus, a Gaussian characteristic function, and therefore any Gaussian state, can be completely specified by 2n 2 + n real parameters.”

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For large temperatures, the FWHH of C(s,T) remains small although larger than C0(s); hence in the high-temperature limit the accuracy of the Markovian master equation stabilizes to a value only a little worse than for T=0.

“For large temperatures, the FWHH of C(s, T ) remains small although now larger than C 0 (s); hence it is expected that in the limit of high temperatures, the accuracy of the Markovian master equation stabilizes to a value only a little worse than for T = 0.”

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Due to the finite number of oscillators in the bath, the simulation is only valid inside a bounded time scale free of back-action, and revivals appear for times t < τ_R, where τ_R is the recurrence time of the bath.

“Due to the finite number of oscillators in the bath, we can only simulate inside a bounded time scale free of the back-action of the bath.This produces revivals in the visualized dynamical quantities for times t < τ R , where τ R is the recurrence time of the bath.”

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The coupling to the bath in the damped harmonic oscillator model is considered in the rotating wave approximation, which is a good description of the real dynamics for small damping.

“Note that the coupling to the bath has been considered in the rotating wave approximation (RWA), which is a good description of the real dynamics for small damping max{g j , j = 1, . . ., M} (e.g. in the weak coupling limit) 5 .”

• cites [5] sourceNot checked yet

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The system-environment state factorization assumption for all times is an effective model and does not make sense as the physical state of the system for all time.

“This assumption has to be understood as an effective model for arriving at equation ( 7) without the use of projection operator techniques; however, it does not make sense to assume that the physical state of the system is really a factorization for all time.”

• AssumptionThe authors state this as an assumption: The whole state factorizes as ρ(t) ≈ ρ_S(t) ⊗ ρ_th at any time.

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The contribution of the correlations to the motion becomes less important as T → 0, counteracting the problem of the infinite width.

“So one may also expect that the contribution of these correlations to the motion becomes less important as T → 0, in such a way that the problem of the infinite width can be counteracted, and this is indeed what seems to happen.”

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The partial trace of a Gaussian state remains Gaussian.

“At the level of covariance matrices, we simply discard elements corresponding to variances including any operators in R 2 , and so the partial trace of a Gaussian state will itself remain Gaussian.”

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The off-diagonal terms of the matrices K(E) and K(A) do not contribute much, β ∼ β, and the rest of the coefficients become approximately equal to those in (58); this happens only under these conditions.

“That is, the offdiagonal terms of the matrices K (E) and K (A) do not contribute much, β ∼ β and the rest of coefficients become approximately equal to those in (58.)Note that this happens only under these conditions.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The rotating wave Hamiltonian is not always an approximation; there are physical systems described physically by that Hamiltonian, for instance when the total number of excitations is preserved.

“In fact, note that the rotating wave Hamiltonian is not always an approximation and there are physical systems that are described physically by that Hamiltonian; for instance, typically in situations where the total number of excitations is preserved.”

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The diagonal terms are also similar for the same reason, so both master equations will be quite similar.

“On the other hand, the diagonal terms are also similar for the same reason, and so both master equations will be quite similar.”

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Although the study focused on harmonic oscillator systems, the proposed method is general and non-harmonic systems should behave similarly with respect to the validity of the equations.

“Although we have focused on harmonic oscillator systems, the proposed method is general and we expect that non-harmonic systems should behave in a similar manner with respect to the validity of the equations.”

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Similar equations for coupled harmonic oscillators have been given previously [42,43], but not in Kossakowski-Lindblad form because those derivations do not take the secular approximation.

“It is worth mentioning that similar equations for coupled harmonic oscillators have been given previously (see for example [42,43]), but not in the Kossakowski-Lindblad form, since in those derivations the secular approximation is not taken.”

• cites [42] Chicone C, 2006, Ordinary Differential Equations with ApplicationsNot checked yet

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• cites [43] Ince E L, 1956, Ordinary Differential EquationsNot checked yet

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In the perturbative series of (B.18), the strength of interaction is not solely dependent on bath coupling because operators A(ν) depend linearly on r/(ω_L-Ω); when this ratio becomes large (r >> 1 or |ω_L-Ω|≈0), the approximation breaks down.

“Before continuing, note that in the perturbative series of (B.18), the 'strength' of the interaction Ṽ (t) is now not solely dependent on the coupling with the bath.This is because the operators A(ν) depend linearly on r w L -, so when this ratio becomes large we expect that the approximation breaks down, i.e. for r 1 or very close to resonance |w L -| ≈ 0.”

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Floquet theory can be used to study this kind of equation and predict whether its solution is periodic.

“This kind of equation can be studied with the well-established Floquet theory (see for example [26,27]); in particular, it is possible to predict if its solution is a periodic function.”

• cites [26] sourceNot checked yet

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• cites [27] sourceNot checked yet

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