Dynamic entanglement in oscillating molecules and potential biological implications
Abstract
We demonstrate that entanglement can persistently recur in an oscillating two-spin molecule that is coupled to a hot and noisy environment, in which no static entanglement can survive. The system represents a non-equilibrium quantum system which, driven through the oscillatory motion, is prevented from reaching its (separable) thermal equilibrium state. Environmental noise, together with the driven motion, plays a constructive role by periodically resetting the system, even though it will destroy entanglement as usual. As a building block, the present simple mechanism supports the perspective that entanglement can exist also in systems which are exposed to a hot environment and to high levels of de-coherence, which we expect e.g. for biological systems. Our results furthermore suggest that entanglement plays a role in the heat exchange between molecular machines and environment. Experimental simulation of our model with trapped ions is within reach of the current state-of-the-art quantum technologies.
The question, to what extent quantum mechanics plays a role in biology, is still far from being well-understood Abbott08 ; Briegel0806 . It seems that classical concepts alone are insufficient for a proper understanding of certain biological processes, and that coherent quantum effects need to be taken into account. It has e.g. long been known that quantum tunneling plays an important role in enzymatic reactions Ball04 ; Tunnelling . Experimental evidence for quantum coherence in the photosynthetic system has recently been reported in Fleming07 . The interplay between the coherent free Hamiltonian and the environment is believed to significantly enhance quantum transport in the Fenna-Matthews-Olson (FMO) protein complex Mohseni0805 ; Rebentrost0807 ; Plenio0807 .
Apart from these isolated instances where quantum coherence seems to help, it is however not clear to what extent biological systems exploit quantum mechanics e.g. to optimize their functionality (beyond the trivial fact that the latter determines, of course, the structure of bio-molecules). Most physicists and biologists are generally skeptical about the question whether genuine quantum features such as entanglement play a broader role in biology. The obvious reason for that viewpoint is that entanglement is very sensitive to noise and requires special conditions to be maintained, in particular very good insulation. Biological systems are anything but - they are wet and hot, and with extremely high levels of noise.
An often ignored fact is, however, that biological systems are also open driven quantum systems, operating far away from thermal equilibrium Briegel0806 . This opens many new possibilities which have not yet been carefully considered. Different from e.g. solid state physics, things in biology move. Protein functions, for example, require conformational motion Frauenfelder99 ; Alberts08 . During such motion, e.g. in the context of protein folding or in isomeric transitions, we have to consider time-dependent quantum interactions (capable of forming e.g. hydrogen or ionic bonds) which are effectively switched on and off while the molecule changes its shape. Since these interactions are accompanied by a substantial amount of noise (e.g. from fluctuating dipole fields from the hydration shell and the bulk solvent), they are usually treated classically. A proper understanding of protein dynamics may however require one to explore the capability of these time dependent interactions and whether they need to be treated quantum mechanically Gilmore08 . It is, for example, not clear whether or not entanglement is generated during these motional processes. A positive answer to this question might reveal novel and subtle aspect of protein dynamics and bio-motoric processes. It would also provide a new twist to the study of nano-bio interfaces, which are just in their infancy.
While it seems reasonable to describe the conformational motion of a bio-molecule classically, it also carries quantum degrees of freedom, such as nuclear spins and electronic states. These give rise to effectively time-dependent quantum interactions, with their strengths depending on the molecular shape, see Fig. 1. We analyze the role of environmental noise and decoherence in such interactions. We find that entanglement can be generated even at room temperature and despite the presence of decoherence, suggesting that the underlying interactions should indeed be described fully quantum mechanically to account for more subtle processes.
As a paradigmatic example, we study the time-dependent interaction of two spins in a thermal and decoherent environment. We may imagine that the spins are attached to some classical backbone structure whose shape changes in time, as drawn schematically in Fig. 2. For simplicity we call such an arrangement a two-spin molecule. We demonstrate that, if the distance between the spins is oscillating, cyclic generation of fresh entanglement can persist, even if no static entanglement can survive. Environmental noise plays thereby both a destructive and constructive role by effectively resetting the system Hartmann06 . The oscillating molecule may be viewed as a molecular machine that exchanges heat with the reservoir. Our results then suggest that entanglement is relevant to the absorption of heat from the environment, which might possibly affect certain biological processes.
In the semi-quantal picture that we have introduced above, the conformational changes lead to classical motion of quantum degrees of freedom as illustrated in Fig. 2. We assume that the two spins are coupled with Ising interaction and that there also exist local electric and/or magnetic fields, both of which are usually position dependent. Thus, the classical molecular motion induces an effective time-dependent Hamiltonian of the form
(1) |
where and are Pauli operators of the th spin, is the interaction strength, and the local level splitting. We emphasize that the subsequent results also hold for more general Hamiltonians, but for simplicity we concentrate here on the Ising interaction. The coupling of the spins to the environment will be described by a master equation of the form
(2) |
where describes the effect of the molecule-environment coupling, and are Lindblad-type generators.
The effect of the environment on the motion of bio-molecules is complex and far from being understood. To demonstrate the essential physics, we first consider a worst-case scenario, where the environment is described by bosonic heat-bath, with each spin being coupled to an independent thermal bath of harmonic oscillators BreuerBook ; Gilmore07 . In the static case, it is well-known that, above a certain temperature, no initial entanglement can persist in such an environment. We will however show that, even under such unfavorable conditions, entanglement can be generated if the particles start oscillating and the system moves out of equilibrium.
If the molecular oscillation is not too fast, in the sense that the adiabatic condition for closed systems is satisfied, the effect of the environment on the oscillating molecule can be described by a master equation of type (2), with implicitly time-dependent Lindblad generators (see also Supplementary Information). As far as the static entanglement is concerned, at every molecular configuration (with fixed and ), the molecule will be driven towards its thermal equilibrium state at temperature . In the following, we adopt the concurrence Wootters98 as the measure of two-qubit entanglement of a state . For a separable (non-entangled) state it vanishes, while for a maximal entangled state it reaches the value 1, i.e. . It can be shown that, if the temperature is above a critical value , no entanglement can survive in any static configuration of the molecule, i.e. . In the following, we will only consider such situations where .
The main question we are asking is this: Can entanglement possibly build-up through the classical motion of the molecule? The answer is affirmative and we demonstrate that entanglement can indeed persistently recur in an oscillating molecule, even if the environment is so hot that the static thermal state is separable for all possible molecular configurations, i.e. .
Let us first present an intuitive explanation. Consider the following simple process: until time , the spins are kept distant (with ) and the molecule is in the thermal equilibrium state, with the fraction of the population in the ground state (). If the local level splitting is sufficiently large such that , will be relatively large compared to the other energy levels. The thermal state will therefore be close to the ground state, which is, in this case, non-entangled. The adiabatic molecular motion from the distant configuration to proximity will transform the eigenstates of into those of ; in particular, the ground state will become entangled as the coupling between the spins increases. This explains, qualitatively, why we may expect entanglement to build up in one run of a conformational change, given that the molecular motion is slow enough to be adiabatic, but at the same time faster than the thermalization process: Driven through the classical motion, the system is so-to-speak “kicked out” of the (separable) thermal equilibrium state, as can be seen in Fig. 3.
The above analysis only suggests that entanglement may appear in one run of a conformational change. However it does not explain how one can expect to see entanglement on a longer time scale, when the environment begins to mix the internal states as the molecule continues to oscillate. It seems that, in the long run, it may (and will) disappear as usual. What we are interested in, however, is the persistent generation of dynamic entanglement, thus an extra mechanism is necessary to refresh the state of the molecule by resetting it back to the initial state. It is intriguing that this role can be played by environmental noise together with oscillatory motion, both of which naturally exist in biological systems without further need for control.
Let us now present the numerical results which we obtained by numerically integrating equation 2. We consider the situation where the spin positions are
(3) |
where are the initial positions, is the amplitude of oscillation, and is the oscillation period. For the local fields we assume Gaussian functions of the spin position as . For the interaction between two spins we assume dipole-dipole coupling with . It can be seen from Fig. 3 that recurrent fresh entanglement appears on the asymptotic cycle. The thermalizing environment is here constructive by re-pumping the population into the ground state.
For biological systems, , the thermal energy is about eV. Energy scales in bio-molecules are typically of the order eV (e.g. for hydrogen bonds or electronic excitation Gilmore08 ; Johnson0708 ) and thus can be several times larger than . The system-bath coupling strength corresponds to a thermalization time of the order of , which compares e.g. with the timescale of (fast) conformational changes within bio-molecules and with relaxation times in the FMO complex Adolphs06 . These numbers seem to be consistent with our conclusion that recurrent entanglement might indeed be found in bio-molecular processes at room temperature.
The oscillating molecule exhibits rather distinct features of non-equilibrium thermodynamics Longpaper , e.g. the entropy does not always increase with time and reach its saturate value. The most intriguing feature is a connection between entanglement and the heat current between the molecule and its environment, which is defined by the energy dissipated via the heat bath as BreuerBook
(4) |
It can be seen from Fig. 3 that, whenever entanglement appears, is always positive, i.e. the molecule tends to absorb heat. This provides some evidence that entanglement is related with the heat exchange between the molecule and its environment. Since the oscillating molecule is not in thermal equilibrium, one cannot adopt the standard definition of temperature. However, by using the spectral temperature as defined in MahlerBook , one finds that the molecule is effectively cooled down through the classical motion, even though the attached thermal bath is always at a fixed higher temperature. This interesting observation is consistent with our common intuition that entanglement appears as the system is cooled down.
The results discussed so far have been obtained by modeling the noisy environment as an Ohmic bosonic heat bath. Clearly, this is a highly idealized model and in any real biological scenario we have far more complicated interactions, e.g. with the surrounding hydration shell and the bulk solvent Fenimore04 ; Gilmore08 . We found similar results also with other decoherence models, based on collision-type interactions of the environment with the system Longpaper . These can be described by Lindblad generators , where is the collision-induced effective relaxation rate and is the mean excitation of a spin in thermal equilibrium. In Fig. 4 we demonstrate the competition between the constructive and destructive effects of environmental noise in such a model, which yields an optimal value for the oscillation period to establish entanglement, see Fig. 4a. For short oscillation periods, efficient thermalization becomes more important: with growing rate , the increase in reset efficiency more than compensates the decrease in efficiency of entanglement generation, see Fig. 4b. In this regime, the net effect of environmental noise is constructive. The dependence of the entanglement on and seems reminiscent of what happens in the very nice example of quantum “stochastic resonance”, which has been described in a quantum-optical context Plenio02 . That phenomenon is, however, fundamentally different from ours, as it involves a bath at zero temperature, and the noise is the main driving force.
Given the complexity of biological systems, how to characterize biological environment is far from been understood. The effect we presented — i.e. the existence of persistent entanglement — is to a very large extent independent of the precise details of the classical movement, and the thermal bath. Of course, the detailed characteristics of the entanglement (how much entanglement, how does it vary with time, etc.) depend on the driving motion and the specific environment, but the very existence of persistent entanglement is generic (extensible to different types of classical motions, spectral density of thermal bath and also to the non-Markovian environment). A number of different models illustrating this generic property are presented in the Supplementary Information.
One can test the feasibility of our model by simulating an oscillating molecule in a noisy environment. Two internal levels of trapped ions can encode an effective two-level system. The system Hamiltonian in the form of Eq.(1) is implementable via state-dependent optical dipole forces Porras04 . The classical oscillation, the essence of which is to introduce a time-dependent Hamiltonian, can be simulated by tuning the interaction strength and the transverse fields, which is achievable e.g. by changing the amplitudes of laser beams Friedenauer08 ; Wineland0805 ; Wunderlich . Decoherence can be simulated by e.g. applying random pulses to induce different decoherence channels. One can also simulate the bosonic bath by engineering the coupling between ions and vacuum modes of the electromagnetic field through laser radiation Poyatos96 . Entanglement can finally be detected by performing quantum state tomography as in RoosPRL . Other implementations are conceivable e.g. using quantum dots mounted on the tips of oscillating cantilevers Bouwmeester .
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I Supplementary Information
This is supporting material for our paper. We present the essential steps in the derivation of the quantum master equation for the oscillating molecule in contact with bosonic heat baths, and the calculation of the concurrence which quantifies the entanglement between the spins. We also describe the spin-gas model as an alternative model for the environment and show that it gives rise to qualitative similar features of reset and recurrent entanglement. Finally, we analyze the competing features of environmental noise for the generation and degradation of entanglement in such a model.
Methods.– To account for the effect of the environment on the oscillating two-spin molecule, we have studied different models, two of which we discuss here. In the first model - the bosonic heat bath - each qubit is coupled to an independent thermal bath of harmonic oscillators. This is a well-known decoherence model and the derivation of the master equation (L2) ^{1}^{1}1References to equations and figures in the Letter start with an “L”, i.e. “eqn. (L2)” means “eqn. (2) in the Letter”. follows standard techniques (see e.g. BreuerBookSI ), with the important difference that in our case the Hamiltonian of the system is time dependent, i.e. its instantaneous energy spectrum changes as the qubits move. This makes the analysis much more complicated, but under certain conditions it can be still be described by an equation of type (L2), albeit with time-dependent Lindblad operators.
In the second model - the spin-gas model Hartmann05SI - each qubit is subject to random, collision-type interactions with a “background gas” of other spin particles. These processes can lead to both local spin exchange and de-phasing. Here we model these processes again by a Lindblad-type master equation Hartmann07SI , but we mention that a numerical treatment including memory effects in the environment, which gives rise to non-Markovian de-coherence, can be given Hartmann05SI .
We calculate the entanglement that is generated during the molecular motion, using the two-qubit measure of concurrence Wootters98SI .
Master equation for the oscillating molecule in contact with bosonic heat baths.– The derivation of the master equation follows standard arguments used in reservoir theory (see e.g. BreuerBookSI ), but with some important modifications due to the time dependence of the problem.
One should point out from the outset that the derivation of the master equation rests on a series of assumptions (including e.g. the Born and Markov approximation and the secular or rotating-wave approximation), neither of which we expect to be very well satisfied in real biological systems.
What the master equation does provide, however, is a dynamical process that exhibits the essential features that we expect to be most relevant in our system of consideration: A process that is disentangling and that leads to de-coherence and thermalization in the subsystem, the two-spin molecule. Whether or not these processes follow, e.g., an exponential decay is not so essential for the main argument.
The total Hamiltonian of the system and the environment can be written in the form
(5) |
where is the time-dependent Hamiltonian (L1) of the two-spin molecule and , describe the oscillator bath and the molecule-bath interactions, respectively. We assume dissipative coupling, in which case the latter can be written in the form
(6) |
where denote the collective bath degrees of freedom that couple to the th spin. The interaction constants have been absorbed in the s.
Within the usual Born-Markov approximation, one obtains an equation of motion for the two-spin molecule which, in the interaction picture, has the form
(7) |
where represents the reduced density matrix of the two-qubit molecule after tracing out the degrees of freedom the thermal bathes.
To perform the secular or rotating wave approximation, we expand the spin operators in (6) into the basis of instantaneous eigenstates with eigenvalues () of the system Hamiltonian , i.e.
(8) | |||||
(9) |
In the first line, the operators describe intra-molecular transitions with frequency and the summation runs over all resonant transition frequencies; in the second line, are the corresponding transition matrix elements and the summation runs over all states with matching energy eigenvalues , . In the interaction picture, which is used in the derivation of the master equation, we then write
(10) |
where is the unitary evolution generated by the system Hamiltonian, , and denotes the time ordering operator.
In a situation where the system’s evolution is adiabatic, i.e. slow enough to avoid energy-changing transitions, the eigenstates will only pick up a dynamic phase (under the coherent evolution of the time-dependent system Hamiltonian) LongpaperSI . The time dependence of the spin operators in the interaction picture then acquires the simple form
(11) |
Upon inserting this into (7) and applying (a generalization of) the secular approximation BreuerBookSI , we obtain a master equation of the form (L2) with implicitly time-dependent Lindblad operators LongpaperSI . The properties of the baths thereby enter through the following Fourier-type transform of the bath correlation functions
(12) |
which is different from the exact Fourier transform obtained in case of a time-independent system Hamiltonian. For independent baths, we only get a contribution for . The Markov approximation assumes that the correlation functions decay fast compared to the relaxation time, which means that their real and imaginary part can essentially be replaced by the delta function and its time derivative , respectively. It is therefore consistent to apply, for small values of , the following approximation for the integral in (12)
(13) |
In summary, we can thus write
(14) |
where the function now depends only on the value of the frequency at the time . Upon transforming back to the Schrödinger picture, in the transition operators are mapped to , and we finally obtain a master equation with the implicitly time-dependent Lindblad generators
(15) |
The index in Eq. (L2) of the Letter runs here over and over the allowed values of . The relevant quantity of the heat bath which enters is its spectral density function. If we assume an Ohmic spectral density with infinite cut-off frequency, we obtain
(16) |
where is the bosonic distribution function at inverse temperature , i.e., . The master equation (L2) with these Lindblad operators was used to calculate the data shown in Fig. L3.
Concurrence and static thermal entanglement.– To measure the dynamic entanglement generated during molecular oscillation, we compare it with the thermal equilibrium state when the molecular configuration is fixed at any distance. In such a “static configuration”, both the Hamiltonian and the Lindblad operators are time independent. Furthermore, the derived quantum master equation is then mixing and the molecule will always be driven to its thermal equilibrium state BreuerBookSI corresponding to the reservoir temperature . For each specific molecular configuration, with fixed spin-spin interaction strength and local electric or magnetic fields , the thermal equilibrium state reads
(17) |
where is the partition function and the matrix representation refers to the computational product basis. The non-zero entries of the above matrix are given by
(18) | |||||
(19) | |||||
(20) | |||||
(21) |
where , , and . To quantify the two-qubit entanglement, there exist various kinds of entanglement measurements. We choose the concurrence Wootters98SI , which is defined as , where the s are the square roots of the eigenvalues of in decreasing order Wootters98SI , with . For the thermal equilibrium state (17), one obtains
(22) |
Using the explicit expressions for one finds that and .
The static thermal entanglement can thus be written as
(23) |
In order to illustrate how the static entanglement changes as the temperature increases, we calculate the first derivative of with respect to . After some straightforward calculations, it can be seen that
(24) |
which means that the static entanglement always decreases as the temperature increases. This is consistent with our intuition that there exists a critical temperature , above which no static entanglement can survive. In other words, for any fixed molecular configuration, entanglement will eventually vanish when the reservoir is too hot. In our Letter, we consider exactly such a situation. The temperature of the environment is so high that the thermal state is separable (non-entangled) at all possible molecular configurations, i.e. .
Spin-gas model for the environment.– The environment of bio-molecular systems is rather complex Frauenfelder09SI and not yet fully understood, and some of its features may not be well-described by a thermal bath model of harmonic oscillators. To check whether the observed effects are robust, we have also considered an alternative model – the so-called spin gas model Hartmann05SI ; Hartmann07SI -- for the environment. In this model, we assume collisions between the molecular spins and other, randomly moving spin-particles that constitute the environment. The collisions induce local energy dissipation (i.e. spin exchange) and de-phasing, which leads to de-coherence and, if left alone, quickly destroys all entanglement between the molecular spins.^{2}^{2}2The spin-gas model should not be confused with the spin-bath model Stamp00SI , which is similar but assumes a static distribution of random couplings between the molecular and the environmental spins.
The spin-gas model of the environment has been described in Refs. Hartmann05SI ; Hartmann07SI . For Ising-type interactions, one can calculate the time evolution of small subsystems exactly, for environments consisting of up to particles, without any approximations. Under certain conditions, one can again derive a master equation by considering the effect of random collisions on a coarse-grained time scale Hartmann07SI . For the present purpose, we will employ such a phenomenological description, but we emphasize that non-markovian and collective effects in the environment can be taken into account Hartmann05SI .
The effect of random collisions leads to Lindblad generators of the form with
(25) |
where are the Pauli ladder operators for a two-level system. The resulting master equation Hartmann07SI describes local energy gain and loss processes (“spin exchange”) with the effective rate , while is related to the temperature and determines the equilibrium distribution of the local excitation Hartmann07SI . Without loss of generality, we may assume that the and . If is larger than a critical value , no static entanglement can exist LongpaperSI , similar as in the case of the bosonic heat bath in the previous section.
Even though this model is quite different from the bosonic heat bath, it has certain features in common, for example, it is disentangling and mixing. Remarkably, we find the same phenomenon as in case of the bosonic heat bath, namely a persistent recurrence of fresh entanglement in a regime where the static entanglement vanishes for all molecular configurations.
In Fig. 5 we compare the evolution of the oscillating molecule for the spin gas model and the bosonic heat bath model. The upper panel of Fig. 5 reproduces the left 3D plot in Fig. L3 (left) of our Letter by projecting it onto two dimensional curves. The lower panel shows the same evolution for the spin gas. It can be seen that the qualitative features are robust: During the first oscillation, entanglement builds up when the spins approach each other, while the evolution subsequently converges towards an asymptotic cycle on which the entanglement periodically recurs. This observation strengthens our claim that this feature is robust and does not seem to depend on the detailed features of the environment.
Competing effects of environmental noise on dynamic entanglement.– Another benefit of the phenomenological spin-gas model is that it allows one to enter the regime of short oscillation periods and to clearly illustrate the competition between the constructive and the destructive effects of the environmental noise.^{3}^{3}3We remark that the task of deriving a simple master equation that is also valid for fast oscillations (i.e. beyond the adiabatic approximation) becomes very hard in the case of the bosonic thermal bath. In Fig. L4, we have plotted the maximal value of entanglement that is assumed on the asymptotic cycle, in the case where (i.e. the equilibrium state is separable for all possible molecular configurations). The left plot displays the maximally achievable entanglement for different oscillation periods. It can be seen that the occurrence of dynamic entanglement strongly depends on the oscillation period; there are competing effects of the environmental noise which give rise to an optimal oscillation period where the effect dynamic entanglement is most pronounced.
To understand the results in Figs. L4 and 6 better, it is worth looking at the detailed time evolution of the entanglement and the ground state population for three typical oscillation periods.
First, consider a very long oscillation period, e.g. . Here,the molecule is almost completely reset by thermalization to equilibrium when the spins are spatially separated (distant configuration), with a large ground state population of up to , see Fig. 6 (blue curve). Since in this regime the coherent evolution of the molecule is adiabatic, the population of the instantaneous eigenstates of the system Hamiltonian remains approximately constant, while the off-diagonal elements remain negligible. When two spins come closer, they start interacting and the ground state becomes entangled (entanglement generation regime). If there were no dissipation, the high population of the ground state alone would be sufficient to generate entanglement. However, while the spins approach each other, the energy separation between the lowest lying levels decreases and the dissipation starts re-populating the levels. This drives the molecular state into the separable regime and diminishes its entanglement, with only little entanglement surviving. For a moderate oscillation period, e.g. (red curve), the dissipation still has enough time to reset the system while it passes through the distant configuration, with a ground state population similar as for as . In the entanglement generation regime, however, the destructive effect of the dissipation is now much smaller than for long oscillation period, which leads overall to more entanglement. Finally, for a very short oscillation period, e.g. , the destructive effect during the entanglement generation regime is even smaller. On the other hand, the reset effect in the distant configuration is greatly suppressed, since the system does not have enough time to thermalize and to repopulate the ground state. Thus, even though the transient entanglement is larger in the first period, as expected, it will diminish in subsequent runs and cannot be sustained on the asymptotic cycle, due to the lack of an effective reset mechanism.
Generic features of persistent dynamic entanglement in noisy environment.— In the main text, we have used a simple model, namely a harmonically oscillating molecule in an Ohmic thermal bath, to demonstrate the essential mechanism for persistent dynamical entanglement to occur in a de-coherent environment where no static entanglement can exist. Here we show that neither the harmonic oscillatory motion nor the Ohmic thermal bath are indispensable for such an effect. The existence of persistent dynamic entanglement is to a very large extent independent of the precise details of the classical motion and thermal environment.
In Fig. 7, we consider a model where the spins move towards (away) from each other with a constant speed, and observe similar results as for the harmonic oscillatory motion. The same effect can also be seen in a scenario of stochastic movements LongpaperSI . In short, the detailed characteristics of the entanglement (how much entanglement, how does it vary with time, etc.) depend on the driving oscillation, but the very existence of persistent entanglement is generic. All that is needed is that the classical motion obeys two conditions: (i) is adiabatically slow but sufficiently fast compared to de-coherence and (ii) it spends long enough time at the far end for thermalization to occur.
Regards the model for the bath, we have so-far used the Ohmic bath (as well as the spin-gas model) as a example. However, our results are also valid for other forms of spectral densities. It can be seen from Fig. 8 that the effect of persistent dynamic entanglement is not restricted to the simple Ohmic bath, but occurs also for the sub-Ohmic and supra-Ohmic bath, i.e. for a spectral density with or . We have also found that the present mechanism works even with the spectral density from the solvent and protein environment McK08SI , e.g. a spectral density in the form of . Finally, by using the numerical method of quasi-adiabatic propagator path integral Mark95aSI ; Mark95bSI , we have extended the results to the non-Markovian environment with finite memory time, and still see the generic effect discussed in the main text. More details will be presented in LongpaperSI .
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