Abstract
We derive an easily computable quantum speed limit (QSL) time bound for open systems whose initial states can be chosen as either pure or mixed states. Moreover, this QSL time is applicable to either Markovian or nonMarkovian dynamics. By using of a hierarchy equation method, we numerically study the QSL time bound in a qubit system interacting with a single broadened cavity mode without rotatingwave, Born and Markovian approximation. By comparing with rotatingwave approximation (RWA) results, we show that the counterrotating terms are helpful to increase evolution speed. The problem of nonMarkovianity is also considered. We find that for nonRWA cases, increasing systembath coupling can not always enhance the nonMarkovianity, which is qualitatively different from the results with RWA. When considering the relation between QSL and nonMarkovianity, we find that for small broadening widths of the cavity mode, nonMarkovianity can increase the evolution speed in either RWA or nonRWA cases, while, for larger broadening widths, it is not true for nonRWA cases.
Introduction
The problem of how fast can a quantum system evolve is of particular interest and has attracted much attention. Quantum mechanics, as a fundamental law of nature, provides ultimate constraints known as quantum speed limits (QSLs) which are virtually at the center of all areas of quantum physics and thus they are of manifold applications, including exploring the physical limits of computation^{1}, providing fundamental limit of precision under quantum metrology^{2,3}, restricting efficiency of quantum optimal control algorithms^{4,5} and providing a minimal time bound to perform the optimal process^{6}. The maximal rate υ of evolution can be described by the QSL time defined as the minimal time τ_{QSL} needed to evolve the initial state (pure or mixed) ρ_{0} to a target state ρ_{t} through a unitary evolution acted by a timeindependent Hamiltonian H, i.e., the shorter time τ_{QSL} means the higher rate υ. The evolution time is lowerbounded by Refs. 7,8,9:
where is the Bures fidelity which can be used to characterize the distance between the two states and is the standard deviation of the initialstate energy. On the other hand, some researchers found that in several cases, the evolution time can be bounded more tightly by the average energy above the ground state, E − E_{0}^{10,11}, namely,
where the initial state ρ_{0} and target state ρ_{t} can be pure or mixed and the energies of the initial state is E = Tr(Hρ_{0}) and the ground state energy is E_{0} = Tr(Hρ_{ground}). Therefore, based on the two results above, people usually define the QSL time as .
Most previous studies focus on unitary dynamics of isolated quantum systems^{7,8,9,10,11,12,13,14,15}, while, all systems are unavoidably coupled to their environments. Therefore, it is necessary to determine a QSL time for open systems^{16,17,18,19}. Recently, Taddei et al.^{16} developed a method to investigate the QSL problem in open systems described by positive nonunitary maps by using of quantum Fisher information for time estimation. While, for the case of the initial mixed states, a Hermitian operator is required to minimize the Fisher information in the enlarged systemenvironment space, which is generally a challenging task. Soon later, del Campo et al.^{17} employed the concept of relative purity to derive an analytical and computable QSL time for open systems undergoing a completely positive and trace preserving evolution. Actually, their bound also easily accounts for the nonMarkovian dynamics. It should be noted that relative entropy can perfectly make a distinction between an initial pure state and its target state, however it may fail to distinguish an initial mixed state to its target state. Because in the latter case, relative purity can reach the value of 1, even though the two states are not completely consistent (see examples in the following section). Recently, Deffner and Lutz^{18} formulated a tight bound on the minimal evolution time of an arbitrarily driven open system and showed that nonMarkovian effects can speed up quantum evolution. However, their time bound is derived from pure initial states and can not be directly applied into the mixed initial states.
Therefore, motivated by the recent studies above, we employ an alternative fidelity definition different from relative entropy in order to derive a new computable QSL time bound, which can easily account for the situations where the initial states are mixed. From this point of view, we emphasize that the new QSL time bound is superior to all the previous bounds in the cases of initial mixed states. On the other hand, the systemenvironment interaction will introduce quantum decoherence, which is one of the most important problem in quantum information processing^{20}. How does the decoherence process affect the QSL time, consequently, affect the evolution speed of the system? In order to give a constructive answer, we study the QSL time problem in a qubit system interacting with a bosonic bath. Moreover, by using of a hierarchy equation method, we try to find the effects of counterrotating terms and nonMarkovianity on the QSL time bound, which are short of study in the previous literatures.
Usually, the description of the dynamics of open systems involves various approximations, such as the Born and Markovian approximation. An effective method that avoids the above two approximations was developed by Tanimura et al.^{21,22,23,24,25}, who established a set of hierarchical equations that includes all orders of systembath interactions. The derivation of the hierarchy equations requires that the timecorrelation function of the bath can be decomposed into a set of exponential functions^{25}. At finite temperature, this requirement is fulfilled if the systembath coupling can be described by a Drude spectrum. The hierarchy equation method has been successfully used in describing quantum dynamics of chemical and biophysical systems^{23,24,25,26,27,28}. On the other hand, the hierarchy equation method is also powerful to study the dynamics of qubit devices at low operating temperature^{29}, when the environment is usually modeled by a Lorentzbroadened cavity mode.
The set of hierarchy equations derived here provides an exact treatment of decoherence and employs neither the rotatingwave, Born, nor Markovian approximations. The hierarchy equation method enables us to deeply explore the effects of the environment on the QSL time bound, which is presented as follows: (i) systembath correlations are here fully accounted during the entire time evolution, which is different from that the correlations are truncated to second order. Highorder correlations are shown^{30} to be very important, even producing a totally different physics; (ii) without weak coupling approximation, the hierarchy equation method is a promising method for studying strong and ultrastrong coupling physics^{31,32}. (iii) Refs. 33 and 34 found that the RWA may lead to faulty results. Especially, recent developments in physical implementation lead to strong coupling between qubit and cavity modes^{31,32}, which requires a careful consideration of the effect of counterrotating terms. Fortunately, the RWA can be avoided in hierarchy equation method; (iv) Markovian approximation is naturally avoided in hierarchy equation method, thus we can consider the effects of nonMarkovianity. Recently more and more attention and interest have been devoted to the study of nonMarkovian processes^{35,36,37,38,39,40,41,42}.
In the following sections, we will first give the QSL time bound derived from a fidelity which is different from the relative purity and Bures fidelity. Secondly, we synoptically introduce hierarchy equation method, then we will consider the QSL time in a qubit system interacting with a broadened cavity mode. The qubitcavity coupling spectrum is described as Lorentz type. The results obtained by hierarchy equations will be compared with those obtained within RWA. The relation between QSL and nonMarkovianity will also be explored.
Results
Derivation of quantum speed limit time
Firstly, we should employ the concept of fidelity as the distance measure of two quantum states. It is well known that the Bures fidelity may be a perfect definition of fidelity^{43}. However, due to the difficulty in the calculation, people try to find alternative definitions of fidelity^{44,45,46}. Among them, we find the definition studied by Wang et al. from the point of Hilbert–Schmidt product for two operators^{46}, has some desirable properties and could be a good distance measure for two states density. The definition reads
where the ρ_{1}, ρ_{2} denote two arbitrary density matrices This fidelity F satisfies Jozsa's four axioms^{43} up to a normalization factor that:

1
F is normalized. The maximum 1 is attained if and only if ρ_{1} = ρ_{2};

2
F is invariant under swapping the two states, i.e., F (ρ_{1}, ρ_{2}) = F (ρ_{2}, ρ_{1});

3
The fidelity is invariant under unitary transformation U on the state space, i.e., F (Uρ_{1}U, Uρ_{2}U) = F (ρ_{1}, ρ_{2});

4
When one of the state is pure, say, ρ_{2} = ψ〉 〈ψ, the fidelity reduces to .
In addition to these advantages, the fidelity (3) is relatively easy to calculate since it only contains the Hilbert–Schmidt inner product and purity. Therefore, it has been well applied in experiment studies such as in NMR system^{47}. Here, it should be compared with the definition of relative purity used in Ref. 17, i.e., . One can easily find that the relative purity ξ fails to satisfy axioms (1) and (2) for general mixed states, e.g., for two different states ( is 2dimensional identity matrix) and ρ_{2} an arbitrary 2dimensional density matrix, one can still obtain ξ (ρ_{1}, ρ_{2}) = 1. From this point of view, relative purity is not suitable to act as a distance measure when the initial state of ρ_{1} is a mixed state and thus it may induce some defects into the derivation of QSL time for this case.
Let us now calculate the changing rate of the fidelity (3). By denoting the initial state as ρ_{0} and the state at time t as ρ_{t}, the derivative of fidelity F (ρ_{0}, ρ_{t}) becomes
The rate above can be further bounded by using CauchySchwarz inequality for operators, i.e., Tr (AB)^{2} ≤ Tr(AA)Tr(BB), then we have
Integrating Eq. (5) over a time period τ leads to the following inequality
where F_{τ} = F (ρ_{0}, ρ_{τ}) denotes the target value of the fidelity in Eq. (3) at time τ and the kernel parameter is defined as
where we retain the time derivative in order that the time limit τ_{QSL} in Eq. (6) can be used to consider either Markovian or nonMarkovian dynamics.
We should note that for the case of a initial pure state ψ (0)〉 under a unitary evolution , then we have , consequently, . Thus the second term in the first line in Eq. (4) equals zero and the coefficient 2 in X_{τ} should be omitted. Furthermore, the minimum time required for the time evolving state ψ (t)〉 to become orthogonal to its initial state ψ (0)〉, i.e., the socalled passage time^{12}, , with (ΔE)^{2} = 〈ψ (0) H^{2} ψ (0)〉 − 〈ψ (0) H ψ (0)〉^{2}, which is consistent with that of Ref. 17. However, for mixed initial states and nonunitary evolutions, our result will be inevitably different from that of Ref. 17.
Moreover, when we take into account a fact that (with d the dimension of ρ_{t}), then we will obtain a looser time bound by substituting into Eq. (6). Consequently, when considering a Lindbladform evolution, we will provide a QSL time bound depending on the initial state and the generators of the dynamical channel similar to the result in Ref. 17.
The systemenvironment model and the hierarchy equation method
Here we consider qubits interacting with a bosonic bath, also known as the spinboson model:
where H_{S} is the free Hamiltonian of the qubit (with ) and here we choose
where σ_{z}_{(x,y)} is the Pauli operator, and
where V is the operator of the qubit and here we choose V = σ_{x}. and b_{k} are the creation and annihilation operators of the bath, while g_{k} is the coupling strength between the qubit and the kth mode of the bath.
The exact dynamics of the system in the interaction picture can be derived as Ref. 25
if the qubit and bath are initially in a separable state, i.e. , where ρ_{B} is the initial state of the bath. In Eq. (11), is the chronological timeordering operator, which orders the operators inside the integral such that the time arguments increase from right to left. Two superoperators are introduced, A ^{×} B ≡ [A, B] = AB − BA and A°B ≡ {A, B} = AB + BA. Also, C^{R} (t_{2} − t_{1}) and C^{I} (t_{2} − t_{1}) are the real and imaginary parts of the bath timecorrelation function
respectively, and
Equation (11) is difficult to solve directly, due to the timeordered integral. An effective method for this problem was developed^{21,22,23,24,25,26,27} by solving a set of hierarchy equations, such as the form of Eq. (17). A key condition in deriving the hierarchy equation is that the correlation function (13) should be decomposed into a sum of exponential functions of time as , with parameters f_{k} and γ_{k} depend on the systembath coupling spectrum and the temperature. Then, the hierarchy equations are obtained by repeatedly taking the derivative of the righthand side.
At finite temperatures, the systembath coupling can be described by the Drude spectrum, however, when we consider qubit devices, which are generally prepared in nearly zerotemperature environments. Then the coupling spectrum between the qubits and cavity modes is usually Lorentz type and the hierarchy method can also be applied^{29}.
Now we consider one qubit interacting with a single mode of the cavity, with transition frequency ω_{0}. Due to the imperfection of the cavity, the single mode is broadened and the qubitcavity coupling spectrum becomes Lorentztype
where λ is the broadening width of the cavity mode which is connected to the bath correlation time . The relaxation time scale on which the state of the system changes is related to γ by τ_{s} = γ^{−1} and γ partly reflects the systembath coupling strength, because when integrating the spectrum J (ω) over the entire region of ω, one will give the effective coupling strength as .
At zero temperature, if the cavity is initially in a vacuum state , the timecorrelation function (13) becomes
which is an exponential form that we need to use for the hierarchy equations. For a limit case, , i.e., , then we have a flat spectrum of Eq. (14) and the correlation tends to δ function that C (t_{2} − t_{1}) → γδ (t_{2} − t_{1}), this is the socalled Markovian limit and the Markovian decay rate γ_{M} = γ.
To derive the hierarchy equation in a convenient form, we further write the real and imaginary parts of the timecorrelation function (15) as
where ν_{k} = λ + (−1)^{k}iω_{0}. Then, following procedures shown in Refs. 21, 25, the hierarchy equations of the qubits are obtained as
where the subscript is a twodimensional index, with integer numbers n_{1(2)} ≥ 0 and . The vectors, , and . We emphasize that with are auxiliary operators introduced only for the sake of computing, they are not density matrices and are all set to be zero at t = 0. The hierarchy equations are a set of linear differential equations and can be solved by using the RungeKutta method. The contributions of the bath to the dynamics of the system, including both dissipation and Lamb shift, are fully contained in the hierarchy equation (17). The Lamb shift term, which is related to the imaginary part of the bath correlation function, can be written explicitly in the common nonMarkovian equations. Since the real and imaginary parts of the bath correlation function are taken into considered here, the effects of the Lamb shift exist in the hierarchy equations, although not in an explicit form.
For numerical computations, the hierarchy equation (17) must be truncated for large enough . We can increase the hierarchy order until the results of ρ_{S}(t) converge. The terminator of the hierarchy equation is
where we dropped the deeper auxiliary operators . The numerical results in this paper were all tested and converged and the density matrix ρ_{S}(t) is positive. The detailed derivation of Eq. (17) can be found in Refs. 21, 29.
Numerical results
In Fig. 1, for a pure initial state, i.e., , where 1〉, 0〉 denote the eigenstates of the Pauli operator σ_{z}, we plot τ_{QSL} of Eq. (6) versus parameter γ by using hierarchy equation method (γ and λ are in units of ω_{0}, which is omitted in the following for simplicity). The systembath interaction Hamiltonian is , so nonRWA case is considered here. The broadeningwidth parameter λ = 0.2 and the actual evolution time is t = 30. For comparison, we also plot which are derived from operator norm, trace norm and HilbertSchmidt norm respectively in Ref. 18. For a Hermitian operator A, if its singular values are μ_{i}, the operator norm is given by the largest singular value , the trace norm is equal to the sum and the HilbertSchmidt norm is defined as . Subsequently, the QSL time can be obtained by substituting into Eq. (6). From numerical results, one can find that corresponding to the highest curve is the tightest bound. And our QSL time τ_{QSL} presents the lowest values in most region of γ, except for small values of γ it gives tighter bound than . Despite the fact that our bound is not tight, it presents similar behavior with the tighter bounds. In the sense of the application in mixed state cases, it allows us to explore the effects of the parameters of the driven Hamiltonian and the purity of the initial states on the QSL time in later parts.
In the following figures, we choose a mixed initial state of Wernertype:
where , the parameter 0 ≤ p ≤ 1 and denotes a 2 × 2 identity matrix.
In Fig. 2, Fig. 3 and Fig. 4, we numerically investigate QSL time bound τ_{QSL} versus parameter γ. A reasonable comparison between the solutions within and without RWA will enable us to understand the contribution of the counterrotating terms. The analytical results of the density matrix at arbitrary time t for RWA case is shown in the method section. Moreover, in order to find the relation between QSL time and nonMarkovianity, we also numerically plot the measure of nonMarkovianity M for RWA and nonRWA cases. Physically speaking, nonMarkovian dynamics implies that the distinguishability of the pair of states increases at certain times. This can be interpreted as a flow of information from the environment back to the system, which prevents the coherence information loss of the system and thus helps to distinguish the two states. Therefore, it is nature to consider that whether nonMarkovianity can accelerate the evolution of the system. The definition of the measure M is shown in the method section.
In Fig. 2a, a small broadening width of the cavity mode λ = 0.03 is chosen, then increasing parameter γ means increasing the qubitcavity coupling strength. We find that for a long region of γ from 0 to about 26.46, τ_{QSL} of nonRWA case (dotted black line) is not longer than that of RWA case (blue line with circles), which means the counterrotatingwave term retained in the nonRWA case can reduce the τ_{QSL}, i.e., increase the evolving speed. While, when γ is too large, τ_{QSL} of nonRWA increases quickly and becomes larger than that of RWA case.
Fig. 2b shows the nonMarkovianity measure M versus parameter γ. In RWA case (blue line with circles), M increases with increasing γ, while in nonRWA case (dotted black line), it is obviously different. Due to the counterrotating terms, it is no longer the case that larger γ can induce greater nonMarkovianity. When γ is larger than a certain value, M begins to decline.
If contrasting the two subfigures Fig. 2a and Fig. 2b, we find that increasing nonMarkovianity will decrease the τ_{QSL} in both RWA and nonRWA cases. While, in nonRWA case, the remarkable reducing process of M for larger γ corresponds to the increase of τ_{QSL}. Therefore, in this small λ case, nonMarkovianity directly affects the QSL time bound, i.e., larger nonMarkovianity decreases τ_{QSL}, while, smaller nonMarkovianity will increase τ_{QSL}.
When we choose a larger parameter λ = 0.1 in Fig. 3, one can see some similar behaviors of τ_{QSL} and nonMarkovianity with Fig. 2, e.g., there is also a crossing of τ_{QSL} for RWA and nonRWA cases, while the crossing point occurs at a smaller value of γ where the nonMarkovianity of nonRWA case begins to decrease.
Different phenomena are shown in Fig. 4, when we continue to enlarge the width to λ = 0.6, there no longer exists the crossing of τ_{QSL} in the two cases of RWA and nonRWA. Instead, τ_{QSL} of nonRWA is always lower than that of RWA case in the presented region of γ. Different dependence behaviors of τ_{QSL} on M are also found for the nonRWA case. When the nonMarkovianity decreases (dotted black line in Fig. 4b), τ_{QSL} (dotted black line in Fig. 4a) also decreases, which is quite different from the former results in Fig. 2 and Fig. 3 where λ is smaller.
Therefore, if someone draws the conclusions that nonMarkovianity can decrease QSL time τ_{QSL}^{18}, our results will provide important additions and amendments that with using RWA, nonMarkovianity can indeed decrease QSL time τ_{QSL}. However, without using RWA, only in the condition of small broadeningwidth parameter λ, increasing nonMarkovianity can depress τ_{QSL}. On the contrary, decreasing nonMarkovianity will increase τ_{QSL}. However, if the width λ is large enough, the dependence of τ_{QSL} on nonMarkovianity will change. Instead, nonMarkovianity reaches the maximum followed by that τ_{QSL} also gets its maximum.
On the other hand, from Fig. 2, Fig. 3 and Fig. 4, one can see the effects of the broadeningwidth parameter λ. If averaging τ_{QSL} over the presented region of γ, one can see that the QSL time in Fig. 4a is larger than that in Fig. 2a and Fig. 3a. Correspondingly, the averaged value of nonMarkovianity M in Fig. 4b is obviously smaller than those of Fig. 2b and Fig. 3b, which implies that increasing the cavity mode width, i.e., enhancing the damping rate of the mode, will reduce the nonMarkovianity and enlarge the τ_{QSL} and thus decelerate the evolution. We should note that the difference between RWA and nonRWA cases demonstrated by τ_{QSL} and M becomes more and more evident when the two parameters γ and λ increase. It is to say, when the time scales of the system and the bath become small, the counterrotating terms will play an important role. Then, the double excitations followed by the virtual exchanges of energy, which are introduced by the counterrotating terms, may prevent the information backow from the environment and thus weaken the nonMarkovianity.
In Fig. 5, we plot τ_{QSL} versus γ for different initial states of Eq. (19) with different mixed coefficients p. Since the purity is defined as . Thus, larger value of p corresponds to higher purity, namely closer to a pure state. Surprisingly, the nonRWA case in Fig. 5a shows some crossings of τ_{QSL}, namely, it is not always a truth that pure state may induce lower QSL time bound than mixed states. Obviously, for larger values of γ, mixed state such as p = 0.1 which is far from pure state, can bring much lower QSL time. But for RWA case, there will not be the strange phenomena any more.
Discussion
We have derived a computable QSL time bound which can be easily applied in the open systems of mixed initial states undergoing nonMarkovian dynamics. By making use of the hierarchy equation method, we considered a qubit system coupled to a broadened cavity mode. We have found that the counterrotating terms (in nonRWA case) can be helpful to decrease QSL times, i.e., accelerate quantum evolution. In nonRWA case, for narrow broadeningwidth λ of the cavity mode, properly enlarging the qubitcavity coupling can decrease QSL time, however, too strong coupling will cause a quickly increasing process of QSL time. While, for wider λs, there exists a maximal QSL time, after that, QSL time decreases monotonically with increasing qubitcavity coupling strength.
On the other hand, in nonRWA case, nonMarkovianity exhibits quite different behavior from RWA case. Too strong qubitcavity coupling may weaken nonMarkovianity. Our results have also demonstrated the close relationship between QSL time and nonMarkovianity. Especially for narrow broadeningwidths of the cavity mode, increasing nonMarkovianity helps to shorten QSL time (for both RWA and nonRWA), while weaker nonMarkovianity may increase QSL time (for nonRWA). While, enlarging broadeningwidth λ will weaken nonMarkovianity and increase QSL time. We also considered initial states with different purity and found that in nonRWA case, the mixed state with lower purity can also lead to shorter QSL time when the qubitcavity coupling is strong enough.
Several tighter bounds depend on exact timeevolution of the density matrix. However, if so, people prefer to consider the exact dynamics behaviour of the density rather than the bound. Therefore, the purpose of deriving a bound is to describe the evolution speed even though we have not grasped enough information of the dynamics. Moreover, the simpler the computation of the bound is, the better it will be applied. From this point of view, we found the only practical QSL bound derived for open systems is that of Ref. 17, which only depends on the initial state and the generators in the Lindbladform evolution. Beyond that, our bound is more available in the case of initial mixed states. Certainly, it is still attractive to explore tight and practical QSL bound in open systems and which leaves lots of interesting problems.
Method
Density matrix within RWA
If we make use RWA, the interaction Hamiltonian in the total Hamiltonian (8) becomes
then with the Lorentztype coupling spectrum and the vacuum initial state of the cavity mode, the density matrix for arbitrary time t can be obtained analytically^{20} as
where the timedependent parameter
with .
Measure of nonMarkovianity: Let us give a brief introduction of the measure of nonMarkovianity defined by Breuer et al.^{35}. For a quantum process, the measure is defined as:
where η [t, ρ_{1,2} (0)] denotes the changing rate of the trace distance that
where is the trace distance of the quantum states ρ_{1} and ρ_{2} with the trace norm definition for a operator . The distance D above characterizes the distinguishability between two quantum states and satisfies 0 ≤ D ≤ 1. It has been pointed that all completely positive and tracepreserving (CPT) maps cannot increase the distance D. For example, η ≤ 0 for all dynamical semigroups and all timedependent Markovian processes, while, if there exists a pair of initial states and a certain time interval such that η > 0, then we can say that the nonMarkovianity appears.
It should be noted that the time integration in Eq. (23) is extended over all time intervals (a_{i}, b_{i}) in which η is positive and the maximum is taken over all pairs of initial states. The measure can be rewritten as:
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Acknowledgements
The authors thank Prof. C. P. Sun, X. X. Yi, Yong Li, Jiangbin Gong, Dario Poletti and Dr. XiaoMing Lu and Shengwen Li for valuable discussions. Z.S. acknowledges support from the National Natural Science Foundation of China under Grant No. 11375003, the Zhejiang Natural Science Foundation with Grant No. LZ13A040002, SUTD StartUp Grant 2012045 the, Program for HNUEYT under Grant No. 201101011 and the funds from Hangzhou City for the HangzhouCity Quantum Information and Quantum Optics Innovation Research Team. X.W. acknowledges support from NFRPC through Grant No. 2012CB921602 and the NSFC through Grants No. 11475146 and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China.
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Z.S., J.L. and X.G.W. calculated and analyzed the results. Z.S. and J.M. finished the numerical calculation. Z.S., J.L. and X.G.W. cowrote the paper. All authors reviewed the manuscript and agreed with the submission.
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Sun, Z., Liu, J., Ma, J. et al. Quantum speed limits in open systems: NonMarkovian dynamics without rotatingwave approximation. Sci Rep 5, 8444 (2015). https://doi.org/10.1038/srep08444
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DOI: https://doi.org/10.1038/srep08444
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