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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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33 No basis given · 29 Own result · 27 Own interpretation · 17 Not checked yet · 3 Own method
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“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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“The first vanishes because V 12 (s) commutes with P and QP = 0.”
§ Markovian master equations: a critical study > Additional information
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“Note that to first order in r and α, we again obtain equation (56).”
§ Markovian master equations: a critical study > (B.19)
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“On the one hand, for a relatively small value of β (= 0.1) in comparison with ω, the off-diagonal elements of the matrices K (E) and K (A) in master equation (59) are unimportant in comparison with the diagonals.”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
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“We have analysed two strategies for finding completely positive Markovian master equations for two harmonic oscillators coupled together under the effect of local baths, indicating that both are complementary in their range of validity.”
§ Markovian master equations: a critical study > 5. Conclusions
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“Surprisingly, the divergences between both equations, even for large times, are actually very small (see figure 10).”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
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“However, when off resonance the difference between the three choices is essentially small.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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“Finally, if we neglect the secular approximation, this regime yields r = r 1 + ( ) + iγ ( ) -ω L - (ω L ) + iγ (ω L ) -ω L .(75)”
§ Markovian master equations: a critical study > 4.2. Markovian master equations
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“Here, the worst behaviour is observed for (73), as expected.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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“We studied the validity of each one and pointed out that completely positive dynamics can be obtained even without secular approximation (for this kind of inhomogeneous equations).”
§ Markovian master equations: a critical study > 5. Conclusions
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“In summary, for the case of a driven damped harmonic oscillator, the difference in accuracy between Markovian master equations is generally small.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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“The details of the derivation are reserved for appendix B.3, but in these three cases we find a Markovian master equation with the structure”
§ Markovian master equations: a critical study > 4.2. Markovian master equations
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“whereas when the driving field is far from resonance, |ω L -| 0, we obtain r = r 1 + ( ) + iγ ( ) -ω L .(74)”
§ Markovian master equations: a critical study > 4.2. Markovian master equations
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“For small Rabi frequencies, an argument similar to section 3.2.1 gives simply r = r,(73)”
§ Markovian master equations: a critical study > 4.2. Markovian master equations
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“Moreover, when the temperatures of the local baths is the same, the difference between them is quite small.”
§ Markovian master equations: a critical study > 5. Conclusions
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“In the derivation in appendix B.3, it is clear why in this case the series diverges for | -ω L | ∼ 0.”
§ Markovian master equations: a critical study > 4.2. Markovian master equations
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“We can diagonalize this quadratic form by means of a rotation to obtain H 12 = + b † 1 b 1 + -b † 2 b 2 , where ± = ( 1 + 2 ) ± 4β 2 + ( 1 -2 ) 2 2 ,”
§ Markovian master equations: a critical study > B.2. Two coupled damped harmonic oscillators, large β
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“In the same spirit, we derived time-inhomogeneous completely positive Markovian master equations for a damped oscillator that is driven by an external semiclassical field.”
§ Markovian master equations: a critical study > 5. Conclusions
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“By construction, the steady state of equation ( 59) is the thermal state of the composed system [3,4]”
§ Markovian master equations: a critical study > 3.3.1. Baths with the same temperature.
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“in fact, we have shown that this is not true in section 4.3”
§ Markovian master equations: a critical study > (B.19)
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“Here we see that both (74) and (75) fail close to resonance, as was expected from the perturbative approach.”
§ Markovian master equations: a critical study > 4.3. Study of the approximations
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“The last equation does not yet warrant complete positivity in the evolution [8], and so we need to perform one final approximation.”
§ Markovian master equations: a critical study > 1.1. Markovian master equations
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- Not checked yetcites [8] source
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“However, in practice, the recurrence time grows very rapidly with the size of the environment and so one can still test the validity of such equations with only a finite, yet still large, environment model, as long as the domain of interest is restricted to early times.”
§ Markovian master equations: a critical study > 1.2. Gaussian states
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“actually we have checked from several simulations that the results turn out to be independent of the number of oscillators as long as the maximum time is less than the recurrence time of the system.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“Note that the shift is independent of the temperature, and although its effect is typically small (e.g.[3,18]), we will not neglect it in our study.”
§ Markovian master equations: a critical study > 2.2. Markovian master equation
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“Due to the linearity in the couplings in H , an initial (global) Gaussian state ρ G will remain Gaussian at all times t, and so we can restrict our attention to the evolution of its covariance matrix,”
§ Markovian master equations: a critical study > 2.1. Exact solution
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“and so the discrepancy with the Markovian master equation is never large.”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“However, they also provide a convenient numerical testing ground as the number of variables needed to model such systems scales polynomially in the number of degrees of freedom.”
§ Markovian master equations: a critical study > 1.2. Gaussian states
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“In addition, note that equation (49) is Gaussian preserving [31], as it is the limit of a linear interaction with an environment and so the total system remains Gaussian, while the partial trace also preserves Gaussianity.”
§ Markovian master equations: a critical study > 2.2. Markovian master equation
What the sources say
- Not checked yetcites [31] source
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“Local versus global master equation with common and separate baths: superiority of the global approach in partial secular approximation”
§ Markovian master equations: a critical study > Abstract
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“If β is sufficiently small to not affect the shift and decay rates, one can expect a Markovian master equation of the form”
§ Markovian master equations: a critical study > 3.2.1. Small intercoupling β.
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“However, if α is small enough, this will happen for values where the contribution of C(s, T ) to the convolution integrals is negligible in comparison with the contribution of C 0 (s), whose FWHH will remain constant and small with respect to τ S .”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“Note that the matrices γ k,ℓ (ν) are positive semidefinite for every ν; this is a consequence of Bochner's theorem [28]; that is, it is easy to check that the correlation functions Tr[ B † k (s)B ℓ ρ th ] are functions of positive type, and γ k,ℓ (ν) are just the Fourier transform of them.”
§ Markovian master equations: a critical study > 1.1. Markovian master equations
What the sources say
- Not checked yetcites [28] source
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“however, the temperatures when this discrepancy is apparent are so small that the contribution from the stimulated process is negligible in comparison with spontaneous emission,”
§ Markovian master equations: a critical study > 2.3.2. Temperature.
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“where each A k can be decomposed into a sum of eigenoperators of the superoperator [H S , •],”
§ Markovian master equations: a critical study > 1.1. Markovian master equations
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“In particular, this implies that they should not be periodic, which requires that the number of degrees of freedom in the environment must be infinite, as otherwise there will be a finite recurrence time.”
§ Markovian master equations: a critical study > 1.1. Markovian master equations
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“In equation ( 12), the terms with different frequencies will oscillate rapidly around zero as long as |ν -ν| α 2 , so in the weak coupling limit these terms vanish to obtain”
§ Markovian master equations: a critical study > 1.1. Markovian master equations
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“The second problem is that there is a lack of rigorous methods for arriving at a completely positive master equation in the Markovian limit when the Hamiltonian is time dependent, with the exception of adiabatic regimes of external perturbations [36].”
§ Markovian master equations: a critical study > 4.2. Markovian master equations
What the sources say
- Not checked yetcites [36] source
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“Similarly to the case of time-independent equations, it can be shown that the family E(t 2 , t 1 ) is completely positive for all (t 2 t 1 ) if and only if L t has the Kossakowski-Lindblad form for any time t [13].”
§ Markovian master equations: a critical study > 4.2. Markovian master equations
What the sources say
- Not checked yetcites [13] source
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“Fortunately in this case, due to the simple periodic time dependence of the Hamiltonian, we will be able to obtain Markovian master equations valid for large (to some degree) Rabi frequencies, even though the complexity of the problem has increased.”
§ Markovian master equations: a critical study > 4.2. Markovian master equations
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“For an arbitrary driving frequency, a Markovian master equation is difficult to obtain as we cannot, in general, make the secular approximation (apart from the perturbative condition |w L -| 0).”
§ Markovian master equations: a critical study > (B.19)
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“So in this master equation the Rabi frequency is renormalized by the effect of the bath.”
§ Markovian master equations: a critical study > (B.19)
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“Next we perform the secular approximation; the cross terms ν = ν in the above, expression, which go as e ±2βti , can be neglected provided that 2β is large in comparison with the inverse of the relaxation rate (β α) and so we obtain”
§ Markovian master equations: a critical study > B.2. Two coupled damped harmonic oscillators, large β
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“This suggests that the general conclusions drawn here are widely applicable to any other settings involving a weak interaction with the environment.”
§ Markovian master equations: a critical study > 5. Conclusions
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“Note that RWA approximation implies β so both normal mode frequencies are positive.”
§ Markovian master equations: a critical study > B.2. Two coupled damped harmonic oscillators, large β
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“We have shown that the system-environment state factorization assumption for all times has to be understood in general as an effective model by deriving the same equation using the projection operator technique.”
§ Markovian master equations: a critical study > 5. Conclusions
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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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“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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“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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“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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“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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“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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“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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“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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“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 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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“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
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
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
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
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]
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
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
plainn = 2- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
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
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
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
“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
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
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
“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.
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
What the paper did — it cites no source for this, so there is none to check. The quote shows it is in the paper.
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.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
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
“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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- The authors state this as an assumption: The whole state factorizes as ρ(t) ≈ ρ_S(t) ⊗ ρ_th at any time.
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
“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.
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
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
“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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
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
“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.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
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
“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
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper may give no reason why it holds. It may state two separate claims as one.
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.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
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
“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)
hedgeddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
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)
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
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
Cites
On the generators of quantum dynamical semigroups
Lindblad, G. (1976) · Communications in Mathematical Physics · doi:10.1007/bf01608499 · Cited by 7,548
Completely positive dynamical semigroups of N -level systems
Gorini, V., Kossakowski, A., & Sudarshan, E. C. G. (1976) · Journal of Mathematical Physics · doi:10.1063/1.522979 · Cited by 3,998
Ensemble Method in the Theory of Irreversibility
Zwanzig, R. W. (1960) · The Journal of Chemical Physics · doi:10.1063/1.1731409 · Cited by 2,366
Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems
Breuer, H.-P., Laine, E.-M., & Piilo, J. (2009) · Physical Review Letters · doi:10.1103/physrevlett.103.210401 · Cited by 1,480
Environment-assisted quantum walks in photosynthetic energy transfer
Mohseni, M., Rebentrost, P., Lloyd, S., & Aspuru-Guzik, A. (2008) · The Journal of Chemical Physics · doi:10.1063/1.3002335 · Cited by 1,189
Statistical Methods in Quantum Optics 1
Carmichael, H. J. (1999) · doi:10.1007/978-3-662-03875-8 · Cited by 1,079
Entanglement and Non-Markovianity of Quantum Evolutions
Rivas, Á., Huelga, S. F., & Plenio, M. B. (2010) · Physical Review Letters · doi:10.1103/physrevlett.105.050403 · Cited by 1,037
Self-citationDephasing-assisted transport: quantum networks and biomolecules
Plenio, M. B., & Huelga, S. F. (2008) · New Journal of Physics · doi:10.1088/1367-2630/10/11/113019 · Cited by 1,002
Self-citationHu, B. L., Paz, J. P., & Zhang, Y. (1992) · Physical Review D · doi:10.1103/physrevd.45.2843 · Cited by 996
Davies, E. B. (1974) · Communications in Mathematical Physics · doi:10.1007/bf01608389 · Cited by 934
Research works printed but not linked: 11
- Shibata F, 1974, Z. Phys., 34, 197
- Breuer H-P, 2002, The Theory of Open Quantum Systems
- Haake F, 1973, Statistical Treatment of Open Systems by Generalized Master Equations (Springer Tracts in Modern Physics vol 66), 98
- Gardiner C W, 2004, Quantum Noise
- Cohen-Tannoudji C, 1992, Atom-Photon Interactions
- Nakajima S, 1958, Prog. Theor. Phys., 20, 984
- Chicone C, 2006, Ordinary Differential Equations with Applications
- Ince E L, 1956, Ordinary Differential Equations
- Reed M, 1980, Methods of Modern Mathematical Physics I
- Bures D, 1969, Trans. Am. Math. Soc., 135, 199
- Apostol T, 1976, Introduction to Analytic Number Theory
Cited by
Sorted by citations
Dynamics of non-Markovian open quantum systems
de Vega, I., & Alonso, D. (2017) · Reviews of Modern Physics · doi:10.1103/revmodphys.89.015001 · Cited by 1,294
Quantum non-Markovianity: characterization, quantification and detection
Rivas, Á., Huelga, S. F., & Plenio, M. B. (2014) · Reports on Progress in Physics · doi:10.1088/0034-4885/77/9/094001 · Cited by 1,035
Self-citationFundamental aspects of steady-state conversion of heat to work at the nanoscale
Benenti, G., Casati, G., Saito, K., & Whitney, R. S. (2017) · Physics Reports · doi:10.1016/j.physrep.2017.05.008 · Cited by 717
Equivalence of Quantum Heat Machines, and Quantum-Thermodynamic Signatures
Uzdin, R., Levy, A., & Kosloff, R. (2015) · Physical Review X · doi:10.1103/physrevx.5.031044 · Cited by 430
The local approach to quantum transport may violate the second law of thermodynamics
Levy, A., & Kosloff, R. (2014) · EPL (Europhysics Letters) · doi:10.1209/0295-5075/107/20004 · Cited by 286
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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