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Markovian master equations: a critical study
Open accessAbstract
We derive Markovian master equations for single and interacting harmonic systems in different scenarios, including strong internal coupling. By comparing the dynamics resulting from the corresponding master equations with numerical simulations of the global system's evolution, we delimit their validity regimes and assess the robustness of the assumptions usually made in the process of deriving the reduced Markovian dynamics. The results of these illustrative examples serve to clarify the general properties of other open quantum system scenarios subject to treatment within a Markovian approximation.
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TL;DR
This paper critically tests Markovian master equations for damped harmonic oscillators by comparing them against exact numerical simulations of Gaussian dynamics. It finds that the global/local master equations are accurate in their respective regimes, but their validity depends on coupling strength, bath temperature, driving, and the approximations used.
Key findings
- For a single damped oscillator, the Markovian master equation provides good accuracy regardless of bath temperature; low-temperature discrepancies from stimulated processes are negligible compared with spontaneous emission, with the correlation function at \(T=0.05\) being \(3.27391\times10^{-7}\) versus \(C_0(s=0)=0.018\).
- For two coupled oscillators with local baths, the weak-intercoupling equation (58) and large-intercoupling equation (59) are complementary; for baths at the same temperature both work well, and in some cases the steady-state fidelity between them exceeds \(99.999\%\).
- There is a small intermediate coupling range, \(\beta \sim 0.01\text{–}0.02\), where neither Markovian master equation achieves high precision, though this range shrinks as bath coupling decreases.
- For a driven damped oscillator, differences between the master equations are generally small; the more elaborate equation (75) works better globally, equation (74) is preferable off resonance, and the simplest equation (73) is more accurate near resonance for small Rabi frequency.
Methods
- The authors derive Markovian master equations using projection-operator/Nakajima–Zwanzig techniques, weak-coupling and secular approximations, and compare them with exact Heisenberg-picture dynamics of harmonic oscillators coupled to finite bosonic baths.
- They simulate Gaussian states via covariance matrices for three systems: a single damped oscillator, two coupled damped oscillators with independent local baths, and a single driven damped oscillator; baths use an Ohmic spectral density \(J(\omega)=\alpha\omega e^{-\omega/\omega_c}\).
- Accuracy is quantified by fidelity/Bures distance between the exact simulated state and the state generated by each Markovian master equation, with representative parameters such as \(\Omega=2\), \(\alpha=1/500\), \(\omega_c=3\), \(\omega_{\max}=20\), and bath oscillator numbers of 350, 700, or 1000.
Limitations
- Stated in the paper: the rotating-wave approximation is used; the Ohmic spectral density with exponential cutoff is a specific choice, and other spectral densities would require re-analysis; finite baths produce revivals after a recurrence time \(\tau_R \propto M\), so simulations are valid only before that time.
- Stated in the paper: the factorized system-environment assumption \(\rho(t)\approx \rho_S(t)\otimes \rho_{\rm th}\) is an effective model rather than a physical description; for time-dependent driven systems, rigorous methods for completely positive Markovian master equations are lacking except in adiabatic regimes, and equations (74) and (75) fail near resonance due to the secular approximation and perturbative series.
- Evident from the text: the numerical tests are restricted to harmonic oscillators and Gaussian states, so the authors’ expectation that non-harmonic systems behave similarly is not directly verified here.
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Claims the paper makes
33 No basis given · 29 Own result · 27 Own interpretation · 17 Not checked yet · 3 Own method
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In the paper
“From these results, one concludes that when modelling a system with multiple baths at different temperatures, equations (58) and (59) are each accurate in their theoretically applicable regimes.”
§ Markovian master equations: a critical study > 3.3. Study of the approximations
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“In summary, the Markovian master equation (49) does not properly describe the stimulated emission/absorption processes (the ones that depend on C(s, T )) for low temperatures;”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“However, for baths at the same temperature, it seems that both equations give good results.”
§ Markovian master equations: a critical study > 3.3. Study of the approximations
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“As a first step, we have plotted the variance of the x coordinate for two different initial states of the system, namely a thermal and a squeezed state (see figure 2).The last plot clearly illustrates the closeness of the results for the Markovian master equation when compared with the effect of the Lamb shift.”
§ Markovian master equations: a critical study > 2.3. Study of the approximations
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“If the oscillators are uncoupled β = 0, it is obvious that the Markovian master equation for their joint density matrix will be a sum of expressions like (49”
§ Markovian master equations: a critical study > 3.2. Markovian master equations
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“As a rough estimation, using the parameters in figure 2, we find that to get a value of the FWHH comparable with τ S ∼ 1/ √ α ∼ 22.4, we need a temperature of at least T ∼ 0.05.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“Because the simulations seem to produce good results for both Markovian master equations when the temperatures of the local baths are the same, regardless of the strength of the intercoupling, it is worth looking at why this happens.”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
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“We see that for the parameters shown on the plot, there is a small range β ∼ 0.01-0.02where neither Markovian master equation obtains a high precision.”
§ Markovian master equations: a critical study > 3.3. Study of the approximations
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“It is manifestly clear that this equation is of the Kossakowski-Lindblad form.”
§ Markovian master equations: a critical study > B.2. Two coupled damped harmonic oscillators, large β
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“We have clarified the possible detrimental effect of low temperatures on the Markovian treatment of a damped oscillator, showing that the Markovian master equation provides good accuracy regardless of the temperature of the bath.”
§ Markovian master equations: a critical study > 5. Conclusions
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“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 .”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“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.”
§ Markovian master equations: a critical study > 2.3.1. Discreteness of the bath.
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“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.”
§ Markovian master equations: a critical study > 3.3. Study of the approximations
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“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
What the sources say
- Not checked yetcites [33] source
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“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).”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“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).”
§ Markovian master equations: a critical study > 3.3. Study of the approximations
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“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).”
§ Markovian master equations: a critical study > Additional information
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“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.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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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%.”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
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“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.”
§ Markovian master equations: a critical study > (B.19)
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“A similar effect can be seen when compared with (74) for larger detunings.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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“However, note that at later times the behaviour of both equations starts to differ, and the steady states are not the same.”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
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“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 .”
§ Markovian master equations: a critical study > B.3. Driven damped harmonic oscillator
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“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”
§ Markovian master equations: a critical study > (B.19)
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- The 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.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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“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.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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“equation (58) is not (although it tends to the thermal state as β → 0, of course).”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
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“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.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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“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)”
§ Markovian master equations: a critical study > Additional information
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- The authors state this as an assumption: Weak coupling implies α << β.
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“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.”
§ Markovian master equations: a critical study > (B.19)
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In the paper
“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.”
§ Markovian master equations: a critical study > B.2. Two coupled damped harmonic oscillators, large β
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“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”
§ Markovian master equations: a critical study > (B.19)
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In the paper
“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.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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“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”
§ Markovian master equations: a critical study > Additional information
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In the paper
“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.”
§ Markovian master equations: a critical study > (B.19)
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In the paper
“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.”
§ Markovian master equations: a critical study > 3.2.2. Large intercoupling β.
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In the paper
“Clearly, J (ω) increases linearly for small values of ω, decays exponentially for large ones and has its maximum at ω = ω c .”
§ Markovian master equations: a critical study > 2. Damped harmonic oscillator
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In the paper
“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.”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
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In the paper
“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.”
§ Markovian master equations: a critical study > 1.2. Gaussian states
What the sources say
- Not checked yetcites [7] source
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In the paper
“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.”
§ Markovian master equations: a critical study > 1.1. Markovian master equations
What the sources say
- Not checked yetcites [6] Breuer H-P, 2002, The Theory of Open Quantum Systems
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In the paper
“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
plainconditional- Asserted: passes — The authors put it forward as true.
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In the paper
“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].”
§ Markovian master equations: a critical study > 1. Introduction
What the sources say
- Not checked yetcites [2]- [6], [8]- [10]
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In the paper
“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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In the paper
“Recently, an exact master equation for two interacting harmonic oscillators subject to a global general environment was derived [23].”
§ Markovian master equations: a critical study > 1. Introduction
What the sources say
- Not checked yetcites [23] Carmichael H J, 1999, Statistical Methods in Quantum Optics. I: Master Equations and Fokker-Planck Equations source
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In the paper
“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.”
§ Markovian master equations: a critical study > 3. Two coupled damped harmonic oscillators
What the sources say
- Not checked yetcites [34] source
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In the paper
“Thus, a Gaussian characteristic function, and therefore any Gaussian state, can be completely specified by 2n 2 + n real parameters.”
§ Markovian master equations: a critical study > 1.2. Gaussian states
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In the paper
“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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In the paper
“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.”
§ Markovian master equations: a critical study > 2.3.1. Discreteness of the bath.
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In the paper
“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 .”
§ Markovian master equations: a critical study > 2. Damped harmonic oscillator
What the sources say
- Not checked yetcites [5] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
plain- Asserted: passes — The authors put it forward as true.
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- The 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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In the paper
“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.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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In the paper
“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.”
§ Markovian master equations: a critical study > 1.2.1. Operations on Gaussian states.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
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In the paper
“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.”
§ Markovian master equations: a critical study > (B.19)
plaincorrelational- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“On the other hand, the diagonal terms are also similar for the same reason, and so both master equations will be quite similar.”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Markovian master equations: a critical study > 5. Conclusions
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None of the sources cited for this claim have been checked yet.
In the paper
“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.”
§ Markovian master equations: a critical study > B.2. Two coupled damped harmonic oscillators, large β
What the sources say
- Not checked yetcites [42] Chicone C, 2006, Ordinary Differential Equations with Applications
- Not checked yetcites [43] Ince E L, 1956, Ordinary Differential Equations
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“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.”
§ Markovian master equations: a critical study > (B.19)
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In the paper
“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.”
§ Markovian master equations: a critical study > (B.19)
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The authors state this as an assumption.
The dynamics of the whole system are given by the Hamiltonian H = H_S + H_E + αV, where V describes the interaction with coupling strength α.
“We then assume that the dynamics of the whole system are given by the Hamiltonian H = H S + H E + αV , where H S and H E are the individual Hamiltonians of the system and environment, respectively, and V describes the interaction between them with coupling strength α.”
The paper relies on this without support.
For the class of interactions considered, Tr_E[Ṽ(t)ρ_E] = 0, which implies PV(t)P = 0.
“For the class of interactions that we are interested in, Tr E [ Ṽ (t)ρ E ] = Tr E [ Ṽ (t)ρ th ] = 0, which implies PV(t)P = 0,(2)”
The authors state this as an assumption.
The initial states of the system and bath are uncorrelated, so the total density operator factorizes as ρ(t0) = ρ_S(t0) ⊗ ρ_th.
“We now assume that the initial states of the system and bath are uncorrelated, so that the total density operator is factorized into ρ(t 0 ) = ρ S (t 0 ) ⊗ ρ th .”
The authors state this as an assumption.
The weak coupling limit is taken by keeping the kernel at lowest order in α (Born approximation).
“We now consider the weak coupling limit, by taking the kernel at lowest order in α, K(t, s) = α 2 PV(t)QV(s)P + O(α 3 ),(6)”
The authors state this as an assumption.
In the limit α → 0, ρ_S changes very slowly and can be taken as constant inside a width τ_B around s = 0.
“Moreover, as ρS changes very slowly in the limit α → 0, we can take it as a constant inside width τ B around s = 0, where Tr E [ Ṽ (t), [ Ṽ (ts), ρ B ]] is not zero, and so finally we obtain”
The authors state this as an assumption.
The whole state factorizes as ρ(t) ≈ ρ_S(t) ⊗ ρ_th at any time.
“assuming that the whole state factorizes as ρ(t) ≈ ρ S (t) ⊗ ρ th at any time [3,10,18,20].”
The authors state this as an assumption.
The strength of coupling to each bath is identical.
“for simplicity we have assumed that the strength of the coupling to each bath is identical (the reader will note afterwards that this is not a crucial assumption).”
The authors state this as an assumption.
The initial state is factorized, Qρ(t0)=0.
“again assuming an initial factorized state (Qρ(t 0 ) = 0)”
The authors state this as an assumption.
The detuning is large enough |ω_L-Ω| >> α, |ω_L-Ω| >> 2αr to make the secular approximation.
“Next we assume that the detuning is large enough |ω L -| α, |ω L -| 2 αr in order to make the secular approximation”
The authors state this as an assumption.
Weak coupling implies α << β.
“weak coupling implies α β”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“The only domain where the global approach breaks down is on resonance, when ω h = ω c and < κ.At this point, the secular approximation is not justified since the two Bohr frequencies ω ± are not well separated”
From The local approach to quantum transport may violate the second law of thermodynamics (2014) · cites this paper as [21]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Markovian master equations: a critical study > Additional information
“J (ω) = j g 2 j δ(ω j -ω) ≈ αωe -ω/ω c .”
§ Markovian master equations: a critical study > Additional information
“j g 2 j ≈ α ω max 0 ωe -ω/2ω c dω”
Citations
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Stochastic quantum thermodynamics, entropy production, and transport properties of a bosonic system
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Atomic switch for control of heat transfer in coupled cavities
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Negative temperature is cool for cooling
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Metadata
- Authors
- Rivas, Á., K Plato, A. D., Huelga, S. F., & B Plenio, M.
- Affiliations
- Universität Ulm (DE)
- Venue
- New Journal of Physics
- Publisher
- IOP Publishing
- Published
- 2010-11-16
- Volume
- 12
- Issue
- 11
- Pages
- 113032
- ISSN
- 1367-2630 (ISSN-L)
- OpenAlex
- W2001389172
- Semantic Scholar
- 9a679fe415a52cb7f30ccd3c64c9f7d649c1643d
- Cited by
- 258
- References
- 60
- Citations by year
- 2026: 10 · 2025: 19 · 2024: 16 · 2023: 48 · 2022: 23 · 2021: 22 · 2020: 24 · 2019: 21 · 2018: 17 · 2017: 10 · 2016: 10 · 2015: 6 · 2014: 12 · 2013: 11 · 2012: 8
- FWCI
- 4.54
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- cc-by
- Locations
- New Journal of Physics (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Engineering and Physical Sciences Research Council
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Keywords
- Markovian master equation, open quantum systems, reduced dynamics, harmonic systems, validity regimes, system-bath coupling, quantum dissipation
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