Abstract
Quantum coherence is one of the primary nonclassical features of quantum systems. While protocols such as the LeggettGarg inequality (LGI) and quantum tomography can be used to test for the existence of quantum coherence and dynamics in a given system, unambiguously detecting inherent “quantumness” still faces serious obstacles in terms of experimental feasibility and efficiency, particularly in complex systems. Here we introduce two “quantum witnesses” to efficiently verify quantum coherence and dynamics in the time domain, without the expense and burden of noninvasive measurements or full tomographic processes. Using several physical examples, including quantum transport in solidstate nanostructures and in biological organisms, we show that these quantum witnesses are robust and have a much finer resolution in their detection window than the LGI has. These robust quantum indicators may assist in reducing the experimental overhead in unambiguously verifying quantum coherence in complex systems.
Introduction
Quantum coherence, or superposition, between different states is one of the main features of quantum systems. This distinctive property, coherence, ultimately leads to a variety of other phenomena, e.g., entanglement^{1,2}. It is also thought to be the power behind several “quantum tools”, including quantum information processing^{3}, metrology^{4}, transport^{5} and recently, some functions in biological organisms^{6} (e.g., efficient energy transport).
Identifying quantum coherence and dynamics in an efficient way, given limited system access, is indispensable for ensuring reliable quantum applications in a variety of contexts. Furthermore, the question of whether quantum coherence can really exist in biological organisms in vivo, e.g., in a photosynthetic complex or in an avian chemical compass, surrounded by a hot and wet environment, has triggered a surge of interest into the relationship between quantum coherence and biological function^{7,8}. In these cases, fullsystem access is often very limited and signatures of quantum coherence are often indirect.
The existing methods for identifying quantum coherent behavior can be generally classified into two types. The first type are based on imposing what can be thought of as a classical constraint^{9}, such as macroscopic realism and noninvasive measurements in the LeggettGarg inequality (LGI)^{10}, or realism and locality in Bell's Inequality. Even though inequalities like the LGI were originally envisaged as a fundamental test of physical theories, a violation of the LGI can also be considered as a tool for classifying the behavior observed in experiment as quantum or classical. However, the LeggettGarg inequality faces severe experimental difficulties when used as such a tool as it requires noninvasive measurements, e.g., via quantum nondemolition (QND) measurement^{11,12}, weak measurement^{13}, or quantumgateassisted ideal noninvasive measurements^{14}. Because of this only a few tests of the LGI have been reported^{14,15,16,17,18}.
The second type of test is based on deduction; do the results of a given experiment sufficiently correspond to the predictions of quantum theory (or classical theory, depending on the approach). Quantum witnesses can be considered as one such test, as they use the knowledge of a quantum state or of some quantum dynamics to determine whether an experimental system possesses quantum properties. Some examples that have been employed elsewhere include witnesses of entanglement^{19,20}, direct measurement of coherence terms of density matrices, or the analysis of process tomography^{21} for nonclassical state evolutions. The experimental realization of this kind of verification usually needs tomographic techniques and then the required experimental resources in terms of measurement settings increases exponentially with the system complexity^{19,21,22}. Moreover, quantum state and process tomography are still difficult to implement in general systems and for general state evolutions, e.g., particularly in systems like charge transport through nanostructures, the transfer of electronic excitations in a photosynthetic complex, or systems where the state space is large.
In this work, we introduce two quantum witnesses to verify quantum coherence and dynamics in the time domain, both of which have various advantages and disadvantages. Both are efficient in the sense that there is no need to perform noninvasive measurements or to use quantum tomography, dramatically reducing the overhead and complexity of unambiguous experimental verification of quantum phenomena.
We apply these quantum witnesses to five examples: (1) electronpair tunnelling in a Cooperpair box and coherent evolution of singletransmon qubit, (2) charge transport through double quantum dots, (3) nonequilibrium energy transfer in a photosynthetic pigmentprotein complex, (4) vacuum Rabi oscillation in lossy cavities and (5) coherent rotations of photonic qubits. Furthermore, as we will illustrate in these examples, our quantum witnesses possess a finer detection resolution than the LGI.
Both witnesses, which we will introduce shortly, involve the following steps: (Figure 1a) first, we prepare the system in a known product state with its environment (or reservoir, here we use both terms interchangeably) ρ_{SR}(0). We then let ρ_{SR}(0) evolve for a period of time t_{0}, to reach the state ρ_{SR}(t_{0}) (during which time one hopes the state has acquired significant coherence due to its internal dynamics). The second step is to implement a quantum witness using a “correlation check” between the state ρ_{SR}(t_{0}) and its state at another later time t ≥ t_{0}, ρ_{SR}(t) (see Figure 1b). The goal of this correlation check is to investigate nonclassical properties in these twotime statestate correlations (see Figure 1c). If the state ρ_{SR}(t_{0}) can be detected then by our quantum witness as having quantum properties, this implies that either the system state ρ_{S}(t_{0}) = Tr_{R}[ρ_{SR}(t_{0})] possesses significant quantum coherence or that the state ρ_{SR}(t_{0}) is an entangled systembath state.
Results
In order to find a signature of quantum dynamics we start by seeking characteristic features of classical dynamics or states^{23}. All separable mixtures of systemreservoir states, with no coherent components, which we call classical states, obey the following relation for their twotime correlations:
See Methods for the proof. Succinctly put, equation (1) implies it is possible to define all future behavior based on only the system's instantaneous expectation values p_{n}(t_{0}). However, most quantum correlation functions also obey this relation under certain measurement conditions. For example, a correlation function constructed from twotime projective measurements has this form as the measurement at t_{0} destroys the coherence in the state at that time. Here Q_{i} is an observable which measures if the system is in the state i. This state is assumed to have a classical meaning (e.g., localized charge state, etc) and the observable is normalized so that its expectation value is directly equal to the probability of observing the system in that state 〈Q_{i}〉 = p_{i}. The propagator Ω_{mn}(t, t_{0}) is the probability of measuring the system in state m at time t given that it was in the state n at time t_{0} (and which in principle depends on the state of the reservoir, so can include classical nonMarkovian correlations, see Methods). Several other recent tests of quantumness^{18,24,25,26,27} rely on imposing Markovianity on Ω_{nm}(t, t_{0}). In our first witness we avoid taking that approach so that we can still distinguish quantum from classical nonMarkovian dynamics. However we will use it in our second witness.
In principle, one could use Eq. (1) to construct a quantum witness of the form:
Where a nonzero result , implies the state at t_{0} can be considered as quantum in that it contains quantum coherence which affects its future evolution. However, as mentioned above, most quantum correlation functions also obey equation (1), which will give . Is it ever possible to observe a nonzero ? In some cases coherence, or “amplitude”, sensitive correlation functions are encountered in quantum optics^{28} and in linearresponse theory^{29}. However, these are typically extracted from spectral functions in the steady state, or put in a symmetrized form, in which case any effect on the correlation function from the initial state coherence may be lost. In all the examples we consider in this work this witness cannot be directly measured, as the initial coherence is of course destroyed by the first (projective) measurement. Fortunately, , via Eq. (1), gives us a way to develop a more generally applicable and valid witness.
Witness 1
Our first practical witness (which is the main result of this work) can be derived from Eq. (1) by including normalization. Noting that all classical systemreservoir states obey,
where d is the number of states n in, or dimensionality of, the system state space, we define our first quantum witness as
If , we can define the state at t_{0} as quantum. Compared with the witness and the tests of the LGI, can always be directly measured and ideal noninvasive measurements are not necessary. In experimental realizations, measuring the populationrelated quantities, or expectation values, 〈Q_{m}(t)〉 and {p_{n}(t_{0})}, is generally more feasible than constructing full correlation functions, particularly in systems which rely on destructive (e.g., fluorescence) measurements. Where correlation functions can be measured with projective measurements, the second term can of course be replaced with Σ_{n} p_{n}(t_{0})Ω_{mn}(t, t_{0}) ≡ Σ_{n}〈Q_{m}(t)Q_{n}(t_{0})〉.
However, determining all the propagators Ω_{mn}(t, t_{0}) with which to construct the witness requires, in principle, that we can prepare the system in each one of it states n exactly (or, alternatively if correlation functions constructed from projective measurements are available, it requires that we measure every possible crosscorrelation Σ_{n}〈Q_{m}(t)Q_{n}(t_{0})〉). In the former case (where we use state preparation) we tradeoff the need to do noninvasive state measurement with the need to perform ideal state preparation. In complex systems it may be difficult to prepare the system in each one of its states to construct these propagators and in some cases we may not even have knowledge of the full statespace of the system.
Importantly, this problem can be easily overcome by noticing that the individual terms in the sum in Eq. (4) are always positive. Thus when constructing the sum we can stop as soon as the witness is violated by this partial summation (i.e., when the terms in the summation together are larger than 〈Q_{m}(t)〉), reducing the experimental overhead substantially (see Figure 4 for a practical example, where we show it is sufficient to include just one term in the sum of Eq. (4)).
Note that with this witness we do not distinguish between just systemcoherence or quantum correlations (entanglement) between system and bath/reservoir (see Methods). In addition, if there are classical correlations between system and reservoir, i.e., classical nonMarkovian effects^{30}, then some additional experimental overhead is needed to eliminate this from giving a “false positive”. If this overhead is ignored this represents a “loophole” in this witness and in some situations may be an obstacle for its unambiguous application. We will discuss this explicitly later with an example of a photosynthetic lightharvesting complex where the system and reservoir are strongly correlated both classically and quantum mechanically.
Witness 2
For our second witness we impose the extra condition that for , for any time interval τ. This assumption restricts us to a widelystudied subset of quantum processes where the systembath/reservoir interaction is Markovian. We will show that, under the assumption that our system lies within this subset, quantum properties can be identified without needing to explicitly measure propagators (i.e., neither exact state initialization or noninvasive measurements are required). The tradeoff in this case is that the witness cannot distinguish certain types of classical dynamics (e.g., classical nonMarkovian), from quantum properties of the system. Still, this witness exceeds the tests proposed in earlier works under the same constraints which still required either noninvasive measurements or state preparation^{18,25}.
This subset of quantum processes can be described as having weak coupling between system and reservoir so that systemreservoir state is always a product state and the bath/reservoir state does not evolve in time, i.e., ρ_{R}(t) = ρ_{R}(0). A large number of systems exist in this regime^{30}, with welldeveloped models such as the master equation under the Born approximation operating within this class (see, e.g.^{30,31,32}). For such cases, we can extend the first witness so that we replace the need to prepare the system state with that of needing to repeatedly measure expectation values (not correlation functions) a number of times that scales linearly with system size. To show this, we consider an extension of Eq. (3) involving a system of d linear equations represented in matrix multiplication form as follows:
where the d × d matrix P_{j} has elements [P_{j}]_{kn} = p_{n}(t_{0[j,k]}) and Ω_{mj} and Q_{mj} are d × 1 column vectors with elements [Ω_{mj}]_{n}_{1} = Ω_{mn}_{[j]}(τ) and [Q _{mj}]_{k}_{1} = 〈Q_{m}(t_{[j,k]})〉, respectively. Here, t_{0[j,k]} and t_{[j,k]} constitute the jth nontrivial timedomain set T_{j}:{t_{0[j,k]}, t_{[j,k]}t_{[j,k]} – t_{0[j,k]} = τ; k = 1, 2, …, d}. For a given time difference τ and a time pair (t_{0[j,k]}, t_{[j,k]}) ∈ T_{j}, one can use the most experimentallyfeasible method of measurement, i.e., invasive measurement, to obtain the information about the state populations, p_{n}(t_{0[j,k]}) and the expectation values 〈Q_{m}(t_{[j,k]})〉.
Given a set of measurement results to sufficiently describe the state populations, the vector Ω_{mj} can be determined by simple algebraic methods. For nonzero determinant det(P_{j}), we have , where is the matrix formed by replacing the nth column of P_{j} by Q_{mj}. For an arbitrary pair of timedomain sets, say T_{j} and , we impose an additional condition (not used in the earlier witnesses) that their propagators should be identical for all classical systems (within the subset described above): . If the system and its environment are classicallycorrelated, i.e., they are not in a product state, this assumption does not hold. Any comparison between Ω_{mj} and can be considered as a quantum witness for this subset, such as the vectorelement comparison:
If and under the assumptions described earlier, we can again assume that some of the (set) of initial states are quantum. Since measuring requires the information about state populations only and can be performed with invasive observations, implementing can be more practical than implementing (2) and (4).
Examples
To illustrate the effectiveness of our witnesses we now present five example systems where they could be applied. For each example we choose which ever witness is more appropriate, given the properties of that system.
Rabi oscillations in superconducting qubits
The oscillations of state populations are commonly thought of as a signature of quantum dynamics. The measurement of these kind of oscillations is widely employed for many experiments. The observation of such oscillations alone, however, is not definitive evidence for the existence of quantum coherent dynamics and can even be mimicked by the solutions of classical autonomous rate equations, e.g., Ref. 33,34.
As a first example of the application of our witnesses we apply (6) to a twolevel system composed of the two lowestenergy states in a singleCooperpair box^{35,36,37}, Figure 2a. We can take n = 1, m = 2, for example together with the designation T_{j}: {t_{0[j,k]} = (k + j – 1)t_{0}, t_{[1,k]} = (k + j – 1)t_{0} + τk = 1, 2} for j = 1, 2, Figure 2b illustrates that the quantum witness detects the presence of quantumness in the Cooperpair tunneling. Since only information about state populations is required, this witness is easy to apply in practice with simple invasive measurements and can be readily applied to the existing experiments in the time domain^{36,37} without any additional experimental overhead.
One can also consider an application of our witnesses to single and multipletransmon qubits coupled to transmission lines in circuit quantum electrodynamics^{38,39}, where qubitstate measurements are performed by monitoring the transmission through the microwave cavity^{39}. For the simplest case of onequbit rotation, the coherent evolution is driven by the Hamiltonian^{38}
where ε(t) is the microwave pulse to induce transitions between qubit states 0〉 and 1〉 with an energy difference ω. Through properly choosing the pulse ε(t), a reliable singlequbit gate, e.g., the Hadamard transformation (H), can be created. Here, we use the quantumprocesstomographybased optimal control theory^{40} to design the microwave pulse for such a gate () with a process fidelity of about 94%. We use the first witness in the form:
to show that the process creates coherent rotations. When setting the input state as 0〉, the value of our witness is about , which certifies the quantumness of .
Quantum transport in quantum dots
Experimentally distinguishing quantum from classical transport through nanostructure remains a critical challenge in studying transport phenomena and designing quantum electronic devices. As mentioned in the introduction, using timedomain methods to verify quantum coherence, such as by testing the LeggettGarg inequality, can be very demanding. We illustrate here how our witnesses are valid in a nonequilibrium transport situation by modelling singleelectron transport through double quantum dots (Figure 3a). Compared with the time periods identified by the LeggettGargtype approach^{25}, the quantum witnesses (Figure 3b) and (Figure 3c) can detect a much larger quantum coherence window. For , we employ the settings T_{j}: {t_{0[j,k]} = [k + c′(j – 1)]t_{0}, t_{[1,k]} = [k + c′(j – 1)]t_{0} + τk = 1, 2 , 3} for j = 1, 2. Here c′ is large such that the whole system is stationary in T_{2}.
Energy transfer in a lightharvesting complex
As an example of the effect of strong interactions with a bath we use a model from biophysics; energy transport in the Fenna–Matthews–Olson (FMO) pigmentprotein complex, where there is thought to be significant systembath entanglement and coherence^{8}. As mentioned earlier, this example enables to discuss the issue of whether classicalcorrelations between system and bath can cause a violation of our first witness (the second witness is not valid in this regime).
In the methods section we impose a classical condition based on an assumption of a class of classical states. States which violate this assumption possess coherences (either in the internal system degrees of freedom, or in the systembath degrees of freedom, i.e., entanglement). However, to prevent classical correlations between system and bath from causing a false positive, the propagators Ω_{mn}(t, t_{0}) in our witness (4), which we construct by preparing the system in one (or more) of its states, must also capture the classical correlations between system and reservoir present at time t_{0}. In the other examples we discuss in this work, this is trivial since the system and bath are always in a product state. However, in systems like the FMO complex we discuss here, this is not the case. Thus to account for these correlations when constructing Ω_{mn}(t, t_{0}) in a general case we must do the following: prepare the systembath product state at t = 0, evolve to time t_{0} and perform a measurement on the system to project it, without preserving coherence, onto one of it states n. We then evolve again, retaining the postmeasurement systembath state and deduce the propagator by measuring the occupation of the state m at final time t. If we can do ideal projective (noncoherence preserving) measurements this accounts for the classical systembath correlation loophole (as long as we can consistently prepare the t = 0 separable systembath state). If we are doing destructive or invasive measurements then we must be able to reprepare the destroyed system state, at time t_{0}, on a time scale faster than the bath/environment dynamics. Since there is no need for measurements on superpositions of basis states, this procedure can be performed without quantum tomography.
We illustrate this with the FMO complex, a sevensite structure used by certain types of bacteria to transfer excitations from a lightharvesting antenna to a reaction center. It has been the focus of a great deal of attention due to experimental observation of apparent “quantum coherent oscillations” at both 77 K and room temperature. To fully capture the nonMarkovian and nonperturbative systembath interactions of this complex system we employ the Hierarchical equations of motion^{7,8}, an exact model (given a bath with a Drude spectral density) valid for both strong systembath coupling and longbath memory time. We use the parameters used by Ishizaki and Fleming in Refs. 7,8 and in Figure 4 we show how this model is detected as quantum by our witness , even at room temperature. We also show, in Figure 4c and 4d, how only partial information about the terms in propagator is needed to find a detection at small times, thus reducing the experimental overhead. In constructing the propagator terms for the sum in Eq. (4) in this case we discard all coherence terms in the physical density matrix but retain the state of the bath, as in^{43}. In this way we account for the state of the bath at time t_{0}, as discussed above. However, accounting for the classical correlations with the reservoir seems beyond the capability of current experiments. We also point out that the full witness detects coherence on timescales greater than t_{0} = 0.3 ps at 77 K, which is a much larger detection window than the LeggettGarg inequality (0.035 ps) for the same parameters^{44}.
Vacuum Rabi oscillation in a lossy cavity
We now consider a Rydberg atom placed in a singlemode cavity which is in resonance with an atomic transition frequency, ω_{0}, for an adjacent pair of circular Rydberg states^{45} e〉 and g〉. Let us consider the case when the cavity field are initially prepared in the excited state e〉 and the vacuum state 0〉_{p}, respectively (denoted by 1〉 = e〉 0〉_{p}). In this case, the atomfield state becomes 2〉 = g〉 1〉_{p} due to spontaneous emission and then periodically oscillates between the states e〉 0〉_{p} and g〉 1〉_{p} at the vacuum Rabi frequency ω_{R}. If the field irreversibly decays due to photon loss out of the cavity, the atomfield stochastically evolves to 3〉 = g〉 0〉_{p} from 2〉. Summarizing the above, the time evolution of the atomfield state ρ can be described by the following master equation^{46}
where is the interaction Hamiltonian of the system. Here κ = ω_{0}/Q and Q is the quality factor of the cavity.
We now use our second witness to detect the vacuumRabi oscillation between the atom and cavity field states. Here we choose the timedomain set as T_{j}: {t_{0[j,k]} = (k + j – 1)t_{0}, t_{[1,k]} = (k + j – 1)t_{0} + τk = 1, 2, 3} for j = 1, 2. Figure 5 shows the value of the witness for vacuumRabi oscillations in a highQ cavity. Using the experimental parameters from^{47}, where , the damped coherent oscillations of the atomcavity state are detected as quantum by our second witness. In comparison, for a lowQ cavity, where 2ω_{R} < ω_{0}/Q, irreversible spontaneous emission out of the cavity will dominate the state evolution. The value of the witness is zero for this case. The measurements on atom states we require to construct the witness are experimentally available by using fieldionization detectors^{45} for selecting atom states e〉 and g〉.
Coherent rotations of photonic quantum bits
Photon polarization states H〉 (horizontal) and V〉 (vertical) have been widely used to achieve linear optical quantum information processing, quantum communication and quantum metrology^{3,48,49}. As a qubit, polarization states can be coherently manipulated by halfwave plates (HWP) and quarterwave plates (QWP). Arbitrary qubit rotations can be performed by using these linear optics elements. Here we will use our first quantum witness to detect the quantum coherence of polarization states created by these rotations. The transformations of HWP and QWP can be represented by the following^{50}:
As a concrete example, one can set a HWP at φ = π/8 to create a photonic Hadamard gate H_{wp}(π/8).
To detect the coherent rotations created by R(φ, θ) = Q_{wp}(θ)H_{wp}(φ), we use the first quantum witness to probe the coherence between states H〉 and V〉. While the witness is originally constructed in the time domain, it can be rephrased in terms of the settings (φ, θ). Assuming that both the wave plates are perfect and there is no photon loss in the birefringent crystals of the wave plates, we have the following correspondences:
and
where ρ_{0} is some initial state created by R. Here m = H and n = V denote the different measurement basis for the horizontal and vertical polarizations. In this example, we set the initial state as ρ_{0} = R^{†}(φ, θ) m〉〈m R(φ, θ) and then the witness becomes
Figure 6 shows this quantum witness for different prepared states ρ_{0}, as a function of the angles θ and φ.
The usual approach to strictly probe the coherent superposition of states H〉 and V〉 is via quantum state tomography^{50}. Compared to such tomographic measurements on single qubit states, which require three local measurement settings, only one setting of a local measurement is now sufficient to implement our first witness.
Discussion
In summary, we have formulated a set of quantum witnesses that allow the efficient detection of quantum coherence, without the restriction of noninvasive measurements. Compared to some of the existing methods, such as the LeggettGarg inequality or employing general quantum tomography, our approach can drastically reduces the overhead and complexity of unambiguous experimental detection of quantum phenomena and has a larger detection window. As illustrated by the five physical examples, these witnesses are robust and can be readily used to explore the presence of quantum coherence in a widerange of complex systems, e.g., transport in nanostructures, biological systems and perhaps even largearrays of qubits used in adiabatic quantum computing^{51}. After this paper went to press, we became aware of this preprint^{52}, which has related results.
Methods
Proof of equation (1)
The quantum twotime statestate correlation 〈Q_{m}(t)Q_{n}(t_{0})〉_{Q} is defined by^{31}:
where ρ_{SR}(t_{0}) is the systemreservoir state and U(τ) is the systemreservoir evolution operator for τ = t – t_{0}. If ρ_{SR}(t_{0}) is a classical state with no coherent components, then we have
where p_{n}(t_{0}) is the probability of measuring the system state n at time t_{0} for the classical mixture ρ_{SR}(t_{0}) and R(t_{0}) is the reservoir state at time t_{0} (which in principle depends on the measurement result Q_{n} if the system and reservoir are classically correlated, i.e., are separable but in a mixture of product states). Then we have
The term describing the system's evolution tr_{R}[U(τ)Q_{n}(0)R(t_{0})U^{†}(τ)] can be described by the operatorsum representation^{21,30}:
where . The the reservoir state is assumed to be R(t_{0}) = Σ_{k} p_{rk} r_{k}〉〈r_{k}. Hence the correlation 〈Q_{m}(t)Q_{n}(t_{0})〉_{Q} for the systemreservoir classical mixture at the time t_{0} is
where is the propagator, i.e., the probability of finding the state m at the time t when the state at an earlier time t_{0} is initialized at n.
The Hierarchy model for FMO
The Hierarchy model was originally developed by Tanimura and Kubo^{41} and has been applied extensively to lightharvesting complexes^{7,8}. We will not give a full description here, but will just summarize the main equation and parameters. It is always assumed that at t = 0 the system and bath are separable and that the bath is in a thermal equilibrium state . The bath is assumed to have a Drude spectral density
where γ_{j} is the “Drude decay constant” and each site j is assumed to have its own independent bath. In addition, λ_{j} is the reorganisation energy and is proportional to the systembath coupling strength. The correlation function for the bath is then given by,
where µ_{j}_{,0} = γ_{j} and when m ≥ 1. The coefficients are
and
Under these assumptions, the Hierarchy equations of motion are given by,
The operator Q_{j} = j〉〈j is the projector on the site j and for FMO there are seven sites, thus N = 7. The Liouvillian L describes the Hamiltonian evolution of the FMO complex. The label n is a set of nonnegative integers uniquely specifying each equation; n = {n_{1}, n_{2}, n_{3}, …, n_{N}} = {{n_{10}, n_{11}, .., n_{1K}}, .., {n_{N}_{0}, n_{N}_{1}, .., n_{NK}}}. The density matrix labelled by n = 0 = {{0, 0, 0.…}} refers to the system density matrix and all others are nonphysical density matrices, termed “auxiliary density matrices”. The density matrices in the equation labelled by indicate that that density matrix is the one defined by increasing or decreasing the integer in the label n, at the position defined by j and m, by 1.
The hierarchy equations must be truncated, which is typically done by truncating the largest total number of terms in a label . This value is termed the tier of the hierarchy. The choice of N_{c} should be determined by checking the convergence of the system dynamics. Here we also use the “IshizakiTanimura boundary condition”^{42};
This can be summed analytically, which for K = 0 gives,
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Acknowledgements
We are grateful to Y. Ota and C. Emary for helpful comments. C.M.L. acknowledges the partial support from the National Science Council, Taiwan, under Grant No. NSC 1012112M006016MY3, No. NSC 1012738M006005 and No. NSC 1032911I006 301 and the National Center for Theoretical Sciences (south). Y.N.C. is partially supported by the National Science Council, Taiwan, under Grant No. NSC 1012628M006003MY3. This research was, in part, supported by the Ministry of Education, Taiwan. The Aim for the Top University Project to the National Cheng Kung University. F.N. acknowledges partial support from the Army Research Office, JSPSRFBR Contract No. 090292114, GrantinAid for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics and Funding Program for Innovative R&D on S&T (FIRST).
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C.M.L. devised the basic model. C.M.L., N.L. and Y.N.C. established the final framework. C.M.L. and N.L. performed calculations and wrote the paper. G.Y.C. attended the discussions. F.N. supervised the project and wrote the paper.
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Li, C., Lambert, N., Chen, Y. et al. Witnessing Quantum Coherence: from solidstate to biological systems. Sci Rep 2, 885 (2012). https://doi.org/10.1038/srep00885
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