shows the heatmap of the chains for the same parameters as Fig. 6.4 except the speed of bosonic excitation $c$ which is doubled. We note that even though $c$ is doubled, the speed of the excitation on the chain remains the same as the rays in both figures 6.4 and 6.5 travel the same distance along the chain in the same time. We recall the result presented in chapter 4, that the propagation speed on the chain is independent of the coupling strength $\alpha$ or the bosonic excitation speed $c$. The propagation speed on the chain depends on the asymptotic hopping energy between the modes of the chain which depends on the cut-off frequency $\omega_c$ of the spectral density which is here held constant [23]. However, for a fixed separation $R$, for $c = 2$ the modes for which the coupling between the chain and the second system site is maximal are twice as close to the origin as the ones for $c = 1$. Hence, for a given distance $R$, it takes half the time for an excitation to travel from the second to the first system site for $c = 2$ than for $c = 1$. The four revivals of eigen-population that we see in Fig. 6.5 correspond to the four rays on the positive chain that come from internal reflections of the initial chain excitation highlighted with arrows. These rays correspond to transmitted parts of bath's excitations bouncing back and forth between the two system sites.

To see the influence of the coupling strength $\alpha$ between the system and the bath, we varied it while keeping a fixed separation $R$ between the system's sites and a fixed speed of phonons $c$. These results are presented in Fig. 6.6 where we can see that increasing the coupling strength sharpens the revivals and brings their peaks closer to $\omega_c t \approx k_c R$. The amplitude of the revivals decrease with the increase of the upper level population prior to the revival.

<figure><img src="image_56.png" /><figcaption>**Figure 6.6:** Comparison of the dynamics of the upper eigenstate at zero temperature for different values of the coupling to the bath $\alpha$. As the coupling increases, the revivals become sharper. The other parameters are held constant at $k_c R = 30$, $c = 1$ and $J = 0.25\omega_c$.</figcaption></figure>

Figure 6.7 shows the coherence in the site basis in the case described by Fig. 6.4 where the initial state of the system is the upper eigenstate. For a degenerate two-level system, the coherences are proportional to the upper eigenstate population. This means that the revivals coincide with a decrease of coherences in absolute value. A decrease of coherences is hence associated with re-localisation.

Another way to show that this revival of eigen-population (re-localisation)
<figure><img src="image_57.png" /><figcaption>**Figure 6.7:** Real and imaginary part of the coherence between the two system sites. The real part is proportional to the upper eigenstate population, hence the revival coincides with a sudden loss of coherence.</figcaption></figure>

is an incoherent mechanism is to look at the evolution of the purity $\lambda = \text{tr}[\rho_S^2]$ of the system state. The purity measures how close state is to a pure state: For $\lambda = 1$, the state is a pure state and for $\lambda = 0.5$ the state of a two level system is a maximal statistical mixture. Figure 6.8 presents the evolution of the purity, and clearly shows that revivals are associated with an increase of mixedness of the system's state.

<figure><img src="image_58.png" /><figcaption>**Figure 6.8:** Purity $\text{tr}[\rho_S^2]$ of the system. The revival corresponds to a loss of purity.</figcaption></figure>

Hence the mechanism behind the revivals can be seen as a partial measurement by the environment on the system's sites that, as a consequence, re-localizes the system's excitation. As the purity is a first order approximation of (one minus) the von Neumann entropy, the decreasing purity at the time of a revival can be seen as an increasing entanglement between the system and its environment. This analysis is consistent with what has been previously said and can be further understood by considering the chain sites that couple the most to the system sites as fragments of the environment that can inform us about the system populations [95]. Indeed, knowing when the chain modes around $n = 20$, in the case where $k_c R = 40$ and $c = 1$, have a gain in population enables us to affirm that there will be a revival at this same
time.

Finite temperature

Using the method presented in chapter 4, we also investigated the finite temperature dynamics of the system. For a large range of values of $\beta$, the system's dynamics stay qualitatively the same except that the steady state population is increased because of thermal fluctuations, as we can see for $\omega_c\beta = 5$ in Fig. 6.9. The peak of the coupling is

<figure><img src="image_59.png" /><figcaption>**Figure 6.9:** (Left) A heatmap of the chain occupation in time showing the propagation of bath excitations along the chains. (Right) Upper eigenstate population. The separation between the two sites is $k_c R = 30$, the speed of sound is $c = 1$, the inverse temperature $\omega_c\beta = 5$ and $\alpha = 0.12$.</figcaption></figure>

at $n = R/c$ and not $R/2c$ as in the zero temperature case, but the propagation speed along the chain is doubled because the support of the extended spectral density is twice as large as the support of the zero-temperature spectral density [78]. The left part of Fig. 6.9 shows the time-frequency diagram for finite temperature for the inverse temperature $\omega_c\beta = 5$ and a separation $k_c R = 30$. For this intermediate temperature, the chain excitation propagate ballistically in way similar to the zero temperature case, except that modes are more populated thanks to thermal fluctuations. Wave-packets emitted from the origin of the chain and the part coupled to the second site interfere when they meet. Hence, we see interference fringes appear when excitations with different phases come together. As in the finite temperature case, when excitations reach the origin of the chain they give rise to a revival of the eigenstate population.
Figure 6.10 shows the upper eigenstate population for increasing values of the temperature. The revivals are still present for moderate temperatures such as $ω_cβ = 5$ but they become barely noticeable for high-temperature, as we can also see in Fig. 6.11. Between $ω_cβ = 5$ and $ω_cβ = 1$ the dynamics of the chains’ modes are the same but the populations are increased by a factor ~ 5. This increased population is a direct consequence of the thermal population. We can see, in Fig. 6.10, that the amplitude of the revival seems to be related to the depth of the plateau reached before $ω_c t ≈ k_c R$. Hence, as the eigen population in this region gets closer to a half, the revival is suppressed.

<figure><img src="image_60.png" /><figcaption>**Figure 6.10:** Upper eigenstate populations for $k_c R = 30$, $c = 1$, $J = 0.25ω_c$ and $α = 0.12$ for several values of the inverse temperature $β$.</figcaption></figure>

For higher temperature, as in Fig. 6.11, the behaviour of the chain is akin to the one we could see for a SBM with a Ohmic spectral density [96] but duplicated on the chain. As they propagate on the chain, excitations leave a trail of populated modes behind them that correspond to the cones we can see on the figure.
<figure><img src="image_61.png" /><figcaption>**Figure 6.11:** (Left) A heatmap of the chain occupation in time showing the propagation of bath excitations along the chains. (Right) Upper eigenstate population. For high-temperature the revival is less pronounced. The separation between the two sites is $k_c R = 30$, the speed of sound is $c = 1$, the inverse temperature $\omega_c\beta = 1$ and $\alpha = 0.12$.</figcaption></figure>

## 6.3 Discussion

Motivated by the ability of biological nanostructures to coordinate (opto)electronic processes through the relaying of environmental (structural) 'signal' motions, we have presented a numerically exact exploration of a model that can describe these highly non-Markovian effects. To do so, we have used the extension of the standard T-TEDOPA techniques presented in previous chapters, in the 1TDVP formulation, to treat the long-range chain couplings that encode information about spatial correlations. In doing so, we have proved that for system-bath problems with spatially correlated interactions, the Hamiltonian matrix product operator will always have a bond dimension proportional to the number of system states, regardless of the range of the interactions. Provided that – as in most models of open systems – the environment is non-interacting, this allows tensor network to be a computationally powerful method for exploring multisite dynamics where non-Markovian environmental feedback could lead to functionally relevant non-equilibrium states and/or transient effects that could materially alter the outcome of a process, if a certain set of events precede it. As our first exploration of this aspect of highly structured nanoscale dissipation, we have shown, with a model composed of two sites, that one of the simplest conceptual forms of correlated environments (plane waves in 1D) supports strong spatio-temporal feedback effects that introduce new timescales into the dissipative dynamics and show clear signs of having stored information about the early time motion,
i.e. after sharp decays, we find sharp revivals. Moreover, we have also
found that periodic behaviour with $T = R/c$ can also be obtained in
which each revival acts as a generator of subsequent revivals, leading
to periodic – but highly anharmonic – energy exchange between the
system states. Finally, we have shown that finite temperatures tend
to broaden and suppress these revival effects, although they visibly
persist for temperatures up to the system energy gap. These results en-
couragingly point to the idea that suitably tailored environments could
be coupled to electronic processes in order to produce well-defined
functional effects at later times and in distant places in the structure.

In the next chapter, building up on our newly acquired knowledge of
non-Markovian signalling we will study how quantum systems can be
dissipatively controlled and we will discuss how this problem relates
to allosteric regulation in proteins.
<figure><img src="image_62.png" /></figure>
Qui cherche trouve toujours. Il ne trouve pas nécessairement ce qu'il cherche, moins encore ce qu'il faut trouver. Mais il trouve quelque chose de nouveau à rapporter à la chose qu'il connaît déjà.

Jacques RANCIÈRE,
Le Maître ignorant

Using the insights on environmental signalling gained in Chapter 6, we now focus on the investigation of the consequences of environmental signalling in dissipative quantum systems. This issue arises from the remarkable advances made in the last decades in the stabilisation and exploitation of quantum effects in man-made nanostructures. This progress opens the possibility for an ever-growing array of technological applications which extract functional advantages from non-classical properties such as coherence and entanglement [97–102] and would constitute the so-called second quantum revolution [16, 28]. However, such quantum properties – particularly when distributed over multiple components/qubits – remain inherently fragile and are easily destroyed by the unavoidable interactions of the functional system states with the macroscopically large number of degrees of freedom that comprise their electromagnetic, electronic, or vibrational environments (as explained in Chapter 2). The theory, simulation and understanding of such open quantum systems is thus central for the development of quantum technologies and, as we shall expose here, could even provide the insights necessary to turn system-environment interactions from a problem into a potent resource. Most of the results presented in this chapter have been made public in Ref. [2].

While a number of strategies have been devised to mitigate the deleterious actions of environmental phenomena, such as relaxation and decoherence, the most common practical approach is to try to insulate the working quantum systems from environmental fluctuations, e.g. by working at low temperatures, or using nanostructuring to suppress the relevant spectrum of environmental excitations and noise [103]. On the other hand, emerging theoretical interest in quantum devices for harvesting and transforming energy has pointed to a number of ways in which dissipative and noisy processes could actually enhance the efficiency of energy transfer and transduction tasks, compared to purely

7.1 Quantum switch . . . . . . . 94
Displaced bath in the chain representation . . . . . . . . . . . 96

7.2 Reorganisation dynamics . 97

7.3 Remote transient activation 100
Sign of the interaction Hamiltonian . . . . . . . . . . . . . . . . 101

7.4 Remote permanent activation . . . . . . . . . . . . . . . 102

7.5 Energy landscape texture 104

7.6 Analogies with allosteric regulation . . . . . . . . . . . . . . 105

7.7 Discussion . . . . . . . . . . . 107

7.A Definition of the reorganisation energy with a toy example . . 108
<figure><img src="image_63.png" /><figcaption>**Figure 7.1:** (a) Spatially distributed systems dissipate energy into their environment and/or cause local deformations that typically propagate and are lost in the bulk medium (red arrows). When systems are packed into nanoscale regions, a significant fraction of environmental excitations will encounter neighbouring systems and influence their dynamics (green arrows), even if the systems are uncoupled (yellow arrow indicates coherent coupling). These interactions are retarded, i.e. depend on the speed of signal propagation and the separation of the systems, providing new time and length scales to their now cooperative dissipative dynamics. Examples in 1D: (b) in photosynthetic reaction centres, pigments are held by a protein scaffold that can dissipatively mediate vibrations and structural reorganisation to coordinate exciton (eh pair) splitting, electron transfer and hole refilling in different locations (separated by 4–5 nm) on timescales from the fs to the µs [92]. (c) Spin-entangled triplet-pairs generated by intramolecular singlet fission interact strongly through the vibrational wave packets of the molecular backbone structure, as in polydiacetylene [121]. (d) A pair of uncoupled quantum dots (QDs) are encapsulated in a nanowire. Exciting one of them thus excites the mechanical modes of the wire. These distortions propagate and could interact with the other QD.</figcaption></figure>

(Fig. 7.1(b)), the existence of extended 1D structural elements (alpha helices) couples charge-induced mechanical/vibrational relaxation at the two functional (donor/acceptor) sites of the system, so that the presence or absence of a charge at a site can modify the activity of the other on relevant timescales [92].

Indeed, it is to be expected that environment-mediated phenomena will be strongly enhanced in low dimensional systems, and especially in 1D, as propagating environmental excitations will necessarily en-counter the other working components. Other, somewhat arbitrarily chosen, real-world 1D systems where strong environmental signalling effects could be expected include entangled triplet exciton dynamics in pi-conjugated polymers and coupled quantum dot emitters grown in nanowires, as shown in Fig. 7.1(c)&(d) [121, 128]. In general, but particularly in molecular matter, injecting ‘system’ excitations causes the local structure to relax to a new equilibrium position, and in doing so key system properties such as energy gaps or couplings to other systems can be strongly modified in the new conformation [39, 40]. As the establishment of this new global conformation must proceed through the propagation of local reorganisation dynamics, dramatic changes at distal locations can be effected at later times [41]. Such a propagating reorganisation could be leveraged to act as a perturbation, a ‘signal’, used to modify the state of a system after a given event has happened, and/or provide the energy to push remote systems out of equilibrium. Alternatively, the same signal could be used, not to control a quantum system, but as a non-destructive sensor of the transitions happening in a system.

To investigate such possibilities, we introduce a fully quantum me-
chanical model that describes how the sudden excitation of a TLS
in a bosonic environment can induce spatiotemporal reorganisation
dynamics capable of *reversibly* triggering and controlling the quantum
<figure><img src="image_64.png" /><figcaption>**Figure 7.2:** (a) Representation of the system. A TLS made of sites *a* and *b*, whose energy difference is $E_b - E_a$, is interacting with the same bosonic environment as the so called "switch" site excited at $t = 0$. This excitation will transiently perturb the environment which will influence the energy levels of the two sites. (b) Heatmap of the energy shift $\Delta E(r,t)$ for an Ohmic spectral density with a hard cutoff induced by an excitation at $r = 0$ with $\lambda$ the bath reorganisation energy, $c = 1$ and $k_c = 1$. The energy shift is composed of a static stabilising contribution centred on the excitation and two destabilising contributions (only $r > 0$ shown), whose amplitudes are half the size, propagating away from the excitation at the speed of sound *c*.</figcaption></figure>

dynamics of a second TLS not directly coupled to the first. Exploiting the numerically exact tensor networks methods that we developed in the previous chapters, we demonstrate that the energy of propagating environmental signals can be harvested by the second TLS to populate metastable excited states that could trigger downstream processes with timing and lifetime information that could be leveraged to match them to other processes. Strikingly, when the TLSs are placed closer to one another, analysis of the complete system-bath wave functions reveals that static, non-local reorganisation can stabilise this activated state, leading to 100% quantum yield and a lifetime that is only limited by the lifetime of the first TLS excited state. We further demonstrate that these ‘transient’ (energy harvesting) and ‘conformational’ (thermodynamic) physics are robust at finite temperatures. We suggest that the rich phenomenology of this remote control OQS model could open up new design concepts for multi-component quantum ‘machines’ and functional materials.

## 7.1 Quantum switch

Figure 7.2(a) shows a schematic representation of the model we shall investigate, which consists of two uncoupled and spatially separated TLSs that couple to the excitations of a common environment. Inspired by 1D mechanically deformable structures coupled to qubits or optoelectronic excitations – e.g. nanomechanical resonators [129–131], organic polymers [132] and the examples of Fig. 7.1 – our environment is taken to be a 1D continuum of spatially extended harmonic oscillators, as would be expected in an extended mechanical structure, and could naturally represent phononic, vibrational, photonic, or magnon baths in specific applications. For simplicity, we take these environment modes to be dispersionless 1D plane waves. The first part of the system is a single site that can be either occupied or unoccupied that we call ‘the switch’ (S). The second part of the system, composed of two
sites $a(b)$ at a distance $r_a(r_b)$ from the switch, is a TLS with an energy gap $E_b - E_a$ and coherent tunnelling coupling $w$. The Hamiltonian describing the system, the environment and their interaction is

$$ \hat{H} = \sum_{i=a,b} E_i \hat{P}_i + w (|a\rangle \langle b| + \text{h.c.}) + \int_{\mathbb{R}} dk \, \hbar \omega_k \hat{a}_k^\dagger \hat{a}_k + \sum_{i=S,a,b} \zeta_i \hat{P}_i \int_{\mathbb{R}} dk \, (g_k^i e^{ikr_i} \hat{a}_k + \text{h.c.}), \quad (7.1) $$

where $\zeta_i = \pm 1$ determines the sign of the coupling, $\hat{P}_i = |i\rangle \langle i|$ projects onto the localised system state $|i\rangle$ in which site $i$ is excited, and $\hat{a}_k^\dagger$ is the bath creation operator for the plane-wave mode of energy $\hbar\omega_k = \hbar c|k|$ with $k$ the wave-vector of the mode and $c$ the speed of sound. The coupling coefficients between the site $i$ and the mode $k$ are $g_k^i e^{ikr_i}$. This coupling induces energy shifts of the system sites that are proportional to the linear displacement of the environmental modes. The sites $a$ and $b$ have the same coupling coefficients $g_k^{a/b} = g_k$ and the ratio of the switch coupling coefficients and the site coupling coefficients is $g_k^S/g_k = \sqrt{k}$. The coupling coefficients define the bath spectral density $J(\omega) = \sum_k |g_k|^2 \delta(\omega - \omega_k)$, which we take to have the widely used Ohmic form $J(\omega) = 2\alpha\omega H(\omega_c - \omega)$ where $\alpha$ is the dimensionless strength of the system-bath coupling, $H(x)$ is the Heaviside function and $\omega_c = c|k_c|$ is the cut-off frequency corresponding to the largest wave-vector $k_c$ of the bath. We use $\omega_c$ as our reference energy scale in all numerical calculations.

We again highlight here that we could, in general, use any spectral function with our techniques.

The initial joint state for the simulation at $T = 0$ is

$$ |\psi(t=0^+)\rangle = |1,1,0\rangle_S \otimes \hat{D}(\{\delta_k\}) |\{0\}\rangle_B \quad (7.2) $$

where $|1, 1, 0\rangle_S$ is the system's state with an excitation on the switch site, an excitation on site $a$ and no excitation on site $b$, $|\{0\}\rangle_B$ is the vacuum state of the bath and $\hat{D}(\{\delta_k\})$ is a multi-mode displacement operator. The displacement comes from allowing the bath to reach equilibrium with the system's state $|0, 1, 0\rangle_S$ where there are no excitations on the switch site and an excitation on site $a$ which is positively coupled to the environment ($\zeta_a = 1$). The relaxed initial state of the environment is a coherent state where each mode $k$ has been displaced by an amount $\delta_k$. Then, at $t = 0^+$, the switch site is excited.

The description of non-vacuum environmental states in the fictitious extended environment used in T-TEDOPA is currently an open problem. Hence at finite temperature the initial system state is unchanged but the environment is described by an undisplaced Gibbs state at inverse temperature $\beta = (k_B T)^{-1}$

$$ \hat{\rho}(t=0^+) = |1, 1, 0\rangle_S \langle 1, 1, 0|_S \otimes \exp(-\beta \hat{H}_B) / Z, \quad (7.3) $$

where $\hat{H}_B = \int_{\mathbb{R}} dk \, \hbar \omega_k \hat{a}_k^\dagger \hat{a}_k$ is the free bath Hamiltonian, and $Z$ is the bath partition function. Nevertheless, we will see that in practice this change of initial condition makes little difference, as the bath relaxes very rapidly to its equilibrium state once the dynamics are initiated by turning on the switch, and this relaxation occurs well before the
propagating signal reaches the distal system.

## Displaced bath in the chain representation

We want the environment to be, at $t = 0$, at equilibrium with an occupied site $a$ and unoccupied switch site and site $b$. One way to obtain this equilibrium state would be to run a first simulation until the desired steady state is reached and use this final state to initialise the next simulation where at $t = 0^+$ the switch site is excited. However, the equilibrium state reached by the environment can be calculated analytically and corresponds to a displaced state. As we will see in the next section, one of the advantages of this approach is that such a state only takes into account the static part of the reorganisation energy. Hence, fewer chain modes are needed compared to the numerical initialisation. The complex amplitudes of the displacement $\delta_k$ of k-modes for a static excitation on site $a$ are given by the solution of the equation of motion of $\langle \hat{a}_k \rangle(t)$. For the system state we described above, we have

$$ \frac{d}{dt} \langle \hat{a}_k \rangle(t) = -i\omega_k \langle \hat{a}_k \rangle(t) - ig_k e^{-ikr_a} . \quad (7.4) $$

We can integrate this differential equation with a bath in its vacuum state as the initial condition $\langle \hat{a}_k \rangle(t = -\infty) = 0$

$$ \langle \hat{a}_k \rangle(0) = -ig_k e^{-ikr_a} \int_{-\infty}^{0} e^{i\omega_k \tau} d\tau . \quad (7.5) $$

Yet, using the identity

$$ \int_{-\infty}^{0} e^{i\omega_k \tau} d\tau = \int_{0}^{+\infty} e^{-i\omega_k \tau} d\tau = \pi\delta(\omega_k) - iP\left(\frac{1}{\omega_k}\right), \quad (7.6) $$

where $P(\cdot)$ is the principal value distribution, and using the fact that $g_k \delta(\omega_k) = 0$ for an Ohmic SD, we finally have

$$ \delta_k \stackrel{\text{def.}}{=} \langle \hat{a}_k \rangle(0) = -g_k e^{-ikr_a} P\left(\frac{1}{\omega_k}\right). \quad (7.7) $$

For any practical purpose because we use an Ohmic spectral density (which is linear in $k$) we can substitute $g_k P(\frac{1}{\omega_k})$ by $g_k/\omega_k$ if we keep in mind that $\delta_{k=0} = 0$. The bath displacement operator $\hat{D}(\{\delta_k\})$ can be written in the chain representation
$$ \hat{D}(\{\delta_k\}) = \exp\left(\int_{\mathbb{R}} (\delta_k \hat{a}_k^\dagger - \delta_k^* \hat{a}_k) dk\right) \qquad (7.8) $$

$$ = \exp\left(\int_{\mathbb{R}^+} (\delta_k \hat{a}_k^\dagger - \delta_k^* \hat{a}_k) dk + \int_{\mathbb{R}^+} (\delta_{-k} \hat{a}_{-k}^\dagger - \delta_{-k}^* \hat{a}_{-k}) dk\right) \qquad (7.9) $$

$$ = \exp\left(\int_{\mathbb{R}^+} (\delta_k \hat{a}_k^\dagger - \delta_k^* \hat{a}_k) dk\right) \exp\left(\int_{\mathbb{R}^+} (\delta_k^* \hat{a}_{-k}^\dagger - \delta_k \hat{a}_{-k}) dk\right) \qquad (7.10) $$

$$ = \prod_n \exp \left( \int_{\mathbb{R}^+} \delta_k U_n(k) \hat{c}_n^\dagger dk - \text{h.c.} \right) \prod_n \exp \left( \int_{\mathbb{R}^+} \delta_k^* U_n(k) \hat{d}_n^\dagger dk - \text{h.c.} \right) \qquad (7.11) $$

$$ = \prod_n \exp \left( \delta_n \hat{c}_n^\dagger - \text{h.c.} \right) \prod_n \exp \left( \delta_n^* \hat{d}_n^\dagger - \text{h.c.} \right) \qquad (7.12) $$

$$ \Rightarrow \hat{D}(\{\delta_k\}) = \hat{D}(\{\delta_n\}) \qquad (7.13) $$

where $\hat{D}(\{\delta_n\})$ is a displacement operator on the chain and $\hat{b}_n^\dagger (\hat{c}_n^\dagger)$ are the chain creation operator originating from positive (negative) $k$-vectors. The complex displacements of chain modes are

$$ \delta_n = \int_{\mathbb{R}^+} \delta_k U_n(k) dk \qquad (7.14) $$

$$ = \int_{\mathbb{R}^+} \frac{-g_k}{\omega_k} e^{-ikr_a} g_k P_n(k) dk \qquad (7.15) $$

$$ \delta_n(r_a) = - \int_{\mathbb{R}^+} \frac{J(k)}{\omega_k} P_n(k) e^{-ikr_a} dk = -2\pi i \mathcal{F} \left[ \frac{J}{\omega} P_n \right] (r_a) . \qquad (7.16) $$

The complex displacement amplitude of a chain mode $n$ is proportional to the Fourier transform (because $J(k)$ is non-zero only for positive $k$) of the reorganisation energy multiplied by the $n^{th}$ order polynomial defining the chain mapping. For an Ohmic spectral density the displacement is proportional to the Fourier transform of the appropriate polynomial, which might be computed analytically. For an Ohmic spectral density, if $r_a = 0$ only the first site of the chain is displaced thanks to the orthogonality of the polynomials. The chain displacement operator $\hat{D}(\{\delta_n\})$ naturally has a MPO representation with bond dimension 1 as can be seen in Eq. (7.12).

## 7.2 Reorganisation dynamics

We first present some physical intuition for the action of the reorganisation dynamics by considering the exactly solvable case of a single excitation on the switch site and occupation of the remote site with a separable system-bath state [35]. The bath part of the interaction Hamiltonian (the last term in Eq. (7.1)) can then be treated as an effective external field that creates a space-dependent shift of the system energies given by

$$ \Delta E(r_{\gamma}, t) = \zeta_{\gamma} \text{tr} \left[ \int_{\mathbb{R}} g_k (\hat{a}_k e^{ikr_{\gamma}} + \text{h.c.}) dk \hat{\rho}_B(t) \right]. \qquad (7.17) $$

$\Delta E(r_{\gamma}, t)$ can be interpreted as work performed by the environment on site $\gamma$ due to its displacement at position $r_{\gamma}$ [113]. From Eq. (7.17),
we have an analytic expression for the energy shift

$$ \Delta E(r_{\gamma}, t) = \zeta_{\gamma} \int_{\mathbb{R}} 2g_k \mathrm{Re} [\langle \hat{a}_k \rangle_B(t) e^{ikr_{\gamma}}] dk, \quad (7.18) $$

and thanks to Ehrenfest theorem given the Hamiltonian in Eq. (7.1)

$$ \langle \hat{a}_k \rangle_B(t) = \frac{g_k}{\omega_k} \sum_{\alpha} e^{-ikr_{\alpha}} \langle \hat{P}_{\alpha} \rangle (e^{-i\omega_k t} - 1) \quad \text{assuming } \langle \hat{P}_{\alpha} \rangle(\tau) \approx \text{cst.} \quad (7.19) $$

Under this assumption, the energy shift can be expressed as

$$ \Delta E(r_{\gamma}, t) = \zeta_{\gamma} \sum_{\alpha} \langle \hat{P}_{\alpha} \rangle \int_{\mathbb{R}} 2 \frac{J(k)}{\omega_k} \left[ \cos(\omega_k t - k(r_{\gamma} - r_{\alpha})) - \cos(k(r_{\gamma} - r_{\alpha})) \right] dk. \quad (7.20) $$

This energy shift can be interpreted for each system site as a wave-packet propagating away from the site with an envelope $\propto J(k)/\omega_k$, plus a static term centred on the site. It can be readily noticed that, because of energy conservation, the energy shift summed over all space at a given time is zero. Trivially, we can see that at $t = 0$ the energy shift vanishes. More generally, for a single localised excitation at $r_\alpha = 0$, for any time $t$ and taking $\zeta_\gamma = +1$

$$ \int_{\mathbb{R}} \Delta E(r,t) dr = \int_{\mathbb{R}} dr \int_{\mathbb{R}} dk 2 \frac{J(\omega_k)}{\omega_k} \left( \cos(\omega_k t - kr) - \cos(kr) \right), \quad (7.21) $$

but $\int_{\mathbb{R}} (\cos(\omega_k t - kr) - \cos(kr)) dr = \delta(k) - \delta(k) = 0$ as $\omega_{k=0} = 0$. Hence, conservation of energy is satisfied. The shape of the wave-packet depends on the bath spectral density $J(k)$. For a SD of the Ohmic family with a soft cutoff the energy shift is

$$ \Delta E(r,t) = 4\alpha ck_c \Gamma(s) \left( \frac{-2 \cos(\arctan(k_c r)s)}{(1+(k_c r)^2)^{s/2}} + \frac{\cos(\arctan(k_c (r-ct))s)}{(1+(k_c (r-ct))^2)^{s/2}} + \frac{\cos(\arctan(k_c (r+ct))s)}{(1+(k_c (r+ct))^2)^{s/2}} \right), \quad (7.22) $$

which reduces in the Ohmic ($s = 1$) case to

$$ \Delta E(r,t) = \lambda \left( \frac{-2}{1+(k_c r)^2} + \frac{1}{1+(k_c (r-ct))^2} + \frac{1}{1+(k_c (r+ct))^2} \right), \quad (7.23) $$

where $\lambda = \int_0^\infty J(\omega)\omega^{-1}d\omega = 4\alpha\omega_c$ is the bath reorganisation energy. The three terms are Lorentz distributions of heights $-2\lambda$, $\lambda$ and $\lambda$ respectively and width $1/k_c$. Figure 7.3 shows the energy shift given by Eq. (7.23) at different times. The first term is the static negative contribution corresponding to twice the reorganisation energy and the two other terms are destabilising (positive) perturbations propagating away from the site. The stationary state is realised by taking $t \to \infty$ and corresponds only to the negative static contribution around the position of the site, i.e. twice the reorganisation energy.

For the hard cut-off case, the results are qualitatively the same as in
<figure><img src="image_65.png" /><figcaption>Figure 7.3: Energy shift $\Delta E(r,t)$ at several times for an Ohmic spectral density with a soft cutoff induced by an excitation at $r = 0$.</figcaption></figure>

the soft cutoff case but with modulation of the wave packet at the wavelength $2\pi/k_c$ due to the Gibbs phenomenon. The energy shift becomes

$$ \Delta E(r,t) = \lambda \left( \frac{-2 \sin(k_c r)}{k_c r} + \frac{\sin(k_c (r-ct))}{k_c (r-ct)} + \frac{\sin(k_c (r+ct))}{k_c (r+ct)} \right). \quad (7.24) $$

Injecting an excitation onto the switch site ($r = 0$) at $t = 0$ would create a time-dependent energy shift of a system at $r_\gamma$, given by

$$ \Delta E(r_\gamma, t) = \frac{-\zeta_\gamma 2\lambda \sin(k_c r_\gamma)}{k_c r_\gamma} + \zeta_\gamma \sum_{\xi=\pm 1} \frac{\lambda \sin(k_c (r_\gamma - \xi ct))}{k_c (r_\gamma - \xi ct)}, \quad (7.25) $$

Figure 7.2(b) shows the time evolution described by Eq. (7.25) for $\zeta_\gamma = 1$; the local relaxation of the environment stabilises the switch excitation on a timescale $\approx \omega_c^{-1}$ (negative energy shift), while two outgoing waves with positive amplitude propagate away from the origin with velocity $c$. These constitute the signals that will act on the distant TLS in the following sections. We note that the positive propagating shifts are a manifestation of conservation of energy under the mechanical distortion of the medium induced by the presence of an excitation on the switch site: in the local bath approximation these contributions are assumed to propagate away rapidly without encountering any other system components, i.e. their energy is essentially lost and the local relaxation is *irreversible*, whereas in our common bath model their interactions with other system components are taken into account (see Fig. 7.1a). At long times only the static contribution centred at the position of the excitation remains (first term of Eq. (7.25)), although it has a non-negligible spatial extension ($\approx k_c^{-1}$) and acts like a potential energy shift on the other sites. Additionally, this potential also shows spatial oscillations that will also play an important role in the full quantum dynamics, below.
## 7.3 Remote transient activation

To study the influence of the transient energy perturbation on the distant TLS at zero temperature, we consider the initial state described above. Site a is positively coupled to the environment ($\zeta_a = 1$) and the environment is in equilibrium with an occupied site a and unoccupied switch site and site b. Throughout, we take $E_b - E_a = 0.5\omega_c$, $w = 0.15\omega_c$ and $\alpha = 0.2$, so that including the reorganisation energy due to the initial relaxation, the total energy gap of the TLS is $E_b - E_a + 2\lambda = 2.1\omega_c$. This is over ten times larger than the coupling $w$, so there is negligible population dynamics in the absence of signals from the switch. At $t = 0^+$, we inject an excitation onto the switch site, hence triggering the reorganisation signal. Figures 7.4(a)-(b) show the dynamics of the population of site b for different switch-site distances $r_a$ and fixed $k_c R = 4$ ($R = r_b - r_a$) both at zero ($\beta = \infty$) and finite ($\beta = 10$) temperatures. All other parameters are held fixed, the coupling signs set to $\zeta_s = \zeta_a = -\zeta_b = 1$ and $\kappa\alpha = 1.2$. The impact of interaction signs on the dynamics will be discussed below. The choice of opposite signs for sites a and b has been made here to enhance the possibility of a population transfer.

When the perturbation generated by the switch-induced reorganisation reaches the system at $t \approx r_a/c$, a sudden transfer of population is initiated that can pump over 50% into the higher-energy site b. Once the perturbation has passed, this population decays back to site a because of downhill energy relaxation. This transfer of population occurs both at $\beta = \infty$ and $\beta \neq \infty$. Figure 7.4(c) shows the evolution at zero temperature of the TLS energy gap as it dramatically closes, as the perturbation raises the energy of site a, and Fig. 7.4(d) is a schematic drawing of the spacing of the sites' energy levels at different moments in time. This transient near-resonance allows coherent transfer of population through an adiabatic transition, creating a metastable excited state whose energy could, for example, be directed towards a desired function. Zero and finite temperatures present the same qualitative dynamics where relaxation of the bath induced by the switch causes a sudden large transfer of population into site b. To reduce computation time and illustrate the general nature of the phenomenon, the finite temperature results shown in Fig. 7.4(b) are for smaller $k_c r_a$ than the zero temperature results. At finite temperature the initial state we use assumes a bath that has not relaxed when there is an excitation on site a; this initial relaxation can be seen in the additional smaller transient population transfer starting at $t = 0^+$ in Fig. 7.4(b). With an unoccupied switch, this initial transfer of population at finite temperature decays back to zero on a timescale of $\omega_c t \sim 40$. The gain of population when the perturbation reaches site a happens at a similar rate in both cases. The available energy of this state has come directly from the work performed on the system by the reorganisation dynamics, and optimising this would be an interesting area for further work. We note a difference in the decay rates between the zero and finite temperature cases. Figure 7.4(b) clearly exhibits exponential decay of the excited state, whereas the highly non-exponential decay of the excited state

We bring to the reader's attention that the dynamics of the TLS energy gap $E_g(t)$ is a joint system-bath observable which would be difficult to extract in a master equation approach.
<figure><img src="image_66.png" /></figure>

**Figure 7.4:** Site *b* population for several distances $r_a$ between the switch and site *a* with $\alpha = 0.2$, $\kappa\alpha = 1.2$, $w = 0.15\omega_c$. (a) $T = 0$, there is no dynamics until $t = r_a/c$. The separation between the sites *a* and *b* is $k_c R = 4$. (b) Finite Temperature $\beta = 10$ presents similar results. (c) Effective energy gap $E_g$ between the two sites as a function of time for $k_c r_a = 104$ at $T = 0$. When the energy perturbation reaches the site it momentarily closes the gap. Three regions are labelled: I before the perturbation reaches the sites; II when the perturbation is closing the gap; and III after the passage of the perturbation. (d) Schematics of the relative spacing between the sites *a* and *b* energy levels (grey lines represent the levels at the previous time).

for zero temperature in Fig. 7.4(a) is suggestive of non-Markovian dissipation. The timing of the population transfer event can be controlled via distance or propagation speed. Indeed, modification of the spectral function may provide a way to tune the excited state lifetime to match its downstream function. Additionally, the philosophy of our model is conserved if one swaps the roles of the switch site and the remote TLS. If one considers the switch site as the system of interest, then the TLS can be seen as a sensor which can monitor transitions of the switch without direct interactions. In that case Figs. 7.4(a)-(b) can be reinterpreted as an indirect way to access the state of the switch site by 'listening' to the environment.

## Sign of the interaction Hamiltonian

To illustrate the influence of the sign of the coupling between system's sites and the environment on the dynamics of the system, let us consider the following choice of parameters: the gap is $E_b - E_a = 0.5\omega_c$, the tunnelling energy is $w = 0.15\omega_c$, the separation between the sites is $k_c R = 10$, the coupling strength is $\alpha = 0.4$, the speed of sound is $c = 1$, $\kappa = 3$ and $\zeta_a = 1$. With such parameters the initial renormalised energy gap is $E_b - (E_a - 8\alpha\omega_c) = 3.7\omega_c \gg w$, hence we expect no dynamics for the populations. However, if the switch site is excited at $t = 0$, the transient energy perturbation should affect the energy of site *a* and site *b* respectively around the times $t = R/c$ and $t = 2R/c$, and could induce some dynamics because the renormalised gap is reduced for a short amount of time. The dynamics of the gap between the two sites will also depend on the sign of $\zeta_b$. A positive sign would mean that the perturbation increases the gap and a negative sign reduces the gap. Figure 7.5 shows the evolution of the population of site *b* for both values of $\zeta_b$. In both cases the dynamics is radically altered as the population grows instead of staying at zero. When $\zeta_b = -1$, two steps are visible and can be clearly associated with the transient energy perturbation. In the opposite case, the population stays constant after the first step. Indeed, the gap becoming larger when the perturbation
reaches site *b* prevents further population transfer. With this example we can see that non-trivial population transfer can be induced by the environment no matter the specific choice of signs in the interaction Hamiltonian.

<figure><img src="image_67.png" /></figure>

**Figure 7.5:** Population of site *b*. Vertical dashed lines are times of the peak of the perturbation reaching site *a* and site *b*. We can see step-like population growth corresponding to the perturbation. The pink dashed line shows the absence of evolution of the population in the absence of the transient perturbation.

## 7.4 Remote permanent activation

A striking change of behaviour is observed when the TLS is brought closer to the switch site, causing the fate of the excited state to become highly sensitive to the switch-TLS distance. Figure 7.6(a) shows the TLS dynamics of the system at zero temperature for smaller $r_a$, where for $k_c r_a = k_c R = 5$, the population is *permanently* transferred from site *a* to site *b* with 100% yield. For $k_c r_a = k_c R = 4,6$ the yield drops to less than 25%. An analogous behaviour can be observed at finite temperature (see Fig. 7.6(b)), where at $\beta = 10$ for $k_c r_a = k_c R = 4,6$ a transient transfer of population to the higher-energy state on site *b* is induced, whereas for $k_c R = 5$ a full transfer is achieved. However the kinetics of the full population transfer are different. Contrary to the $\beta = \infty$ case, at finite temperature the $k_c r_a = k_c R = 4,6$ populations decay to the lower energy eigenstate populations. This is due to the presence of thermal fluctuations that allow the remnant population on site *b* to explore the energy landscape and decay back to the lower energy site *a*. The stability of population transfer at $k_c R = 5$ suggests a lasting thermodynamic change in the energy ordering of the TLS sites, which we have traced to the role of the static shift $\Delta E(r, t \to \infty)$ induced by the switch (Eq. (7.25)). This can be directly visualised by computing $\Delta_{ab}E(x, t) = \sum_{i=a,b} \langle \psi(t) | \zeta_i \hat{P}_i \int_R dk (g_k e^{ikx} \hat{a}_k + \text{h.c.}) | \psi(t) \rangle$. This gives a representation of the evolution of the transient 'energy landscape' perceived by the TLS, and is composed of a large number of two-time two-point correlation functions that can be readily evaluated within our many-body tensor networks simulation framework.

To gain more insight into these effects Fig. 7.7(a) shows a heatmap of $\Delta_{ab}E$ for the full population transfer case ($k_c R = 5$) at $\beta = \infty$. At
<figure><img src="image_68.png" /></figure>

**Figure 7.6:** (a) For $k_c r_a = k_c R = 5$ there is a full population transfer at $\beta = \infty$ – both when the full reorganisation dynamics is taken into account and when only the static part of the landscape is considered – that is not seen for other positions. In the $k_c R = 5$ case, the population decays back to site *a* when the switch site is de-excited at $\omega_c t = 200$ (dashed vertical line and orange background). (b) Finite Temperature $\beta = 10$ also presents a full population transfer.

$t = 0^+$ the environment starts to relax because of the presence of the switch and generates a static negative energy shift at $x = 0$ and two propagating positive energy shifts (only the positive $x$ one is shown) which propagate at the speed of sound. When the right-propagating transient perturbation reaches site *a*, it lifts its energy level and thus initiates population transfer to site *b*. However, once the perturbation passes, the population transfer continues. Fig. 7.7(b) shows a cross section of the heat maps at the final time and the energy landscape in the hypothetical case where the environment has relaxed to an equilibrium state in presence of an excited switch and a populated site *a*. It can be seen that after the switch is fully relaxed, an occupied site *b* now corresponds to the global ground state of the system, and local environmental dissipation will thus drive the system to this state, as illustrated in Fig. 7.7(c).
<figure><img src="image_69.png" /><figcaption>**Figure 7.7:** (a) Energy landscape perceived by sites *a* and *b* (marked by dashed lines) as a function of time and space for *k*<sub>c</sub>*R* = 5, *α* = 0.2, *κ*α = 1.2, and *β* = ∞. (b) Energy shifts for the state a long time after the transient perturbation has passed (i.e. relaxed environment in the presence of the excited switch and site *b*) and the state where the environment reached an equilibrium when the switch is excited and the population of the remote TLS is localised on site *a* (*w* = 0). As the absolute minimum of the energy landscape moves from the location of site *a* to that of site *b*, the population evolves correspondingly. (c) Schematics of the absolute energy levels of sites *a* and *b* for the different configurations.</figcaption></figure>

7.5 Energy landscape texture

Light can be shed on the distance-dependence of this stabilised trans-
fer by also considering the ‘texture’ of the energy shift induced by the
sinc-shaped static part of the switch-generated energy perturbation
(see Fig. 7.8(b)). The switch generates a static negative energy shift
of large amplitude around the origin and several local minima and
maxima that alternate in sign with a wavelength 2π/kc. The further
away from the switch, the lower the amplitude of these extrema (at
kcx = 10, 90% of the amplitude has been lost already). In the kcR = 5
case, site a is close to a maximum of this landscape and is raised in en-
ergy. Site b sits on the second maximum, but because it couples to the
environment with a negative sign, this static contribution corresponds
to a stabilising shift. The energy landscape texture depends on the
nature of the bath spectral density. Sharp decays or super-Ohmicity
for example can be a source of texture. For the parameters used in
our simulations, these static shifts cause the bare energy of site b to
become lower than site a, after the switch has relaxed. Small changes
δR/R < 1 in the position of the sites would not drastically alter the full
population transfer results as the important factor for this process to
happen is that the static contribution is enough to invert the energy
gap. Although the initial relaxation of the environment around site
a still creates a large barrier for population transfer, the static switch
potential now renders the population on site a metastable. Thus, the ac-
tivation of the switch has primed a permanent conformational change
in the environment that drives full population transfer. This is directly
evidenced in Fig. 7.6(a), showing that an excitation initially localised
on site a undergoes spontaneous and complete transfer to site b, when
the environment is initially relaxed in the presence of a populated
switch. This observation has a strong implication, namely that this
transition should be reversible once the switch is depopulated, as is
verified in Fig. 7.6(a) for kcR = 5 when we remove the switch excitation
at ωct = 200. All these transitions can be summarised by looking at the
sites' energy levels for different configurations, as shown in Fig. 7.7(c). Reversibility of transitions is of paramount importance, from a control perspective, as it enables the switching of the state of the TLS using a mechanical medium; our model highlights how spatial variations in the non-local energy shift could be exploited for dissipatively 'locking in' activated states. This reversible 100%-yield population transfer also enables the possibility of information transduction from, for instance, an electromagnetic medium (a laser-pulse used to excite the switch site) to a mechanical medium. An additional way to frame this effect of spatial variation of the reorganisation energy landscape is to consider it as a simple form of environment engineering. The excited switch site precisely located could be considered as part of the environment of the TLS — which would effectively be described with a new spectral density — whose dissipative dynamics now spontaneously populates the TLS site *b*. Hence, one could design a dissipative energy landscape by combining several switch-like systems and placing them at selected positions in space. Finally, in contrast to dynamically activated metastable states, we note that 'permanently activated' states are favoured thermodynamically, i.e. do not harvest energy from the triggering of the switch directly; the work done by the environment has gone into creating the *driving force* for the *dissipative* population transfer. The deterministic movement of, say, charged excitations to new sites could be used to trigger a wide range of chemical and mechanical processes which could provide catalytic coordination or sensing functions.

## 7.6 Analogies with allosteric regulation

Allostery refers to the central regulation process in proteins in which the binding of an effector molecule (ligand) at one site of the protein can have a reversible at-a-distance effect on a distal site: increasing or decreasing its ability to bind (see Fig. 7.8(a)). Despite having been conceptualised almost sixty years ago [133, 134], understanding the biophysical mechanisms behind this long-range coupling remains a highly active topic [135–137]. Hence, numerous structural and thermodynamical models [138–143] to explain this crucial process have been proposed but, even though some give quantitatively accurate predictions, none of these alone provide a convincing explanation for the nature of allosteric signalling. Therefore, allostery is often referred to as "the second secret of life" [144].

The at-a-distance environmentally driven dynamics studied in this chapter presents common features with the biological process of allosteric regulation. Indeed, the simple model presented here exhibits several features that are central to the mechanism of allostery:

* the mechanical nature of the signal,

* the distal control of a transition,

* the reversibility of the transition upon removal of the effector, and

The first secret of life has already been uncovered when it was demonstrated that DNA is the physical support of hereditary information.
**Figure 7.8:** (a) Representation of an allosteric enzyme. The binding of an effector molecule on the allosteric site can change the configuration of a distal binding site. (b) The components of the model presented in this chapter have been relabelled to their analogous components in an allosteric enzyme. The binding of the ligand will transiently perturb the environment which will influence the stability of the binding-site's configurations.

<figure><img src="image_70.png" /></figure>

► the specific spatial arrangement of allosteric and active sites (called allosteric pathways).

Our generic OQS model can be thought of as a basis for describing the effect of a ligand-binding site to induce a transition between two possible enzyme conformations. Hence, one could assign to the different elements of our models counterparts in the prototypical description of enzyme allostery. The switch site could play the role of an allosteric site and the states of the distal TLS could stand for two different binding site conformations. The vibrational environment would then be associated with the rest of the enzyme. Figure 7.8(b) shows how the analogy is constructed. In this analogy, the excitation of the switch corresponds to the binding of an effector at the allosteric site. This binding generates a structural reorganisation of the enzyme which induces a conformational change at the binding site. This conformational change of the whole structure of the enzyme now renders another configuration of the binding site more stable, henceforth changing the binding ability of the site. Reversibility of transitions once the effector molecule unbinds is one of the key dynamical features of allosteric regulation, and our model highlights how spatial variations in the non-local energy shift could be exploited for dissipatively 'locking in' activated states whose lifetimes are now set by the binding time of a ligand. In such a model, the allosteric signal sending information from one site to the other is the reorganisation energy landscape introduced above, and the non-local reorganisation can stabilise this configuration, leading to 100% quantum yield and a lifetime that is only limited by the dwell time of the ligand. This simple model is able to recover the key features of allostery and provide a clear picture of a potential underlying mechanism in terms of reorganisation energy. However, we wish to highlight that the model presented here is not a model describing biological allostery and it cannot make any quantitative predictions in its current formulation about, for instance, chemical reaction rates or the actual arrangement of the allosteric pathways. Such a quantitative model would need additional ingredients, the most straightforward one being a biologically relevant SD that corresponds to a specific enzyme. Another ingredient would be a better description of the vibrational environment (e.g. dispersion, additional dissipation of the vibrational mode themselves...) and its coupling with the binding sites (under which hypothesis is it linear?). Nevertheless, this model of non-Markovian remote control could provide high-level insights that could be relevant for actual biological allostery. I would like to add environmental signalling through transient energy
shifts and background energetic texture to the list of possible elements
of understanding of the fundamental issue of allosteric regulation.

**7.7 Discussion**

Our open quantum model of non-Markovian, environmentally me-
diated signalling, allows us to leverage unavoidable dissipative pro-
cesses to reversibly control the transition of a remote TLS. Using our
numerically exact tensor networks simulation methods, two distinct
classes of controlled process — transient and permanent activations —
have been identified both at zero and finite temperatures, and further
work will explore the efficiency of these mechanisms in single-shot
and cyclic, i.e. engine, operation. Crucially our quantum switch model
provides a natural platform to explore strongly non-classical effects
in remote signalling, and could naturally be extended to multiparticle
systems to explore spatiotemporal entanglement dynamics driven by
dissipative processes. Indeed, this quantum switch model could pro-
vide a reversible and non-invasive means of controlling, entangling
and probing qubits. Physical 1D platforms such as cold atoms, super-
conducting qubit chains and ion traps could also be used as quantum
simulators for such physics. The study of common environment dis-
sipative physics that can give rise to non-local and non-Markovian
effects is crucial in the context of quantum engineering in order to
be able to channel dissipation toward useful functions, and to miti-
gate induced cross-talk errors in multi-component nanodevices. The
possibility, opened by our model, for the environment to be in a su-
perposition of two different displacements of opposite magnitudes
hints toward indefinite causal order of dissipative processes [145, 146].
In another direction, our model could be refined in multiple ways to
describe realistic (bio-)chemical systems. For instance, our present the-
oretical tools could be used to approach structured spectral densities
found by ab initio methods [83, 147], anharmonic effects [148, 149], or
Hamiltonian topologies that account for more complex connectivity
between systems and/or environments [39]. Another perspective for
this model is to provide a new set of processes to analyse the intricate
physical effects happening in biological systems made of complexes
of organic molecules such as allostery [134, 135, 142] or multi-electron
processes [124].
# Appendix

## 7.A Definition of the reorganisation energy with a toy example

Consider a system interacting with a single harmonic oscillator of angular frequency $\omega$ with the following Hamiltonian

$$ \hat{H} = \hat{H}_S + \frac{\hbar\omega}{2}(\hat{X}^2 + \hat{P}^2) + g\hat{S}\hat{X} \qquad (7.26) $$

where $\hat{X}$ and $\hat{P}$ are respectively the dimensionless displacement and dimensionless momentum operators of the harmonic oscillator, $\hat{S}$ is a diagonal system operator, and $g$ is the coupling strength between the system and the oscillator. The linear coupling term between the system and the oscillator can be removed by changing $\hat{X} \to \tilde{X} = \hat{X} + \hat{A}$, i.e. by changing the equilibrium position of the oscillator. Hence the harmonic oscillator's Hamiltonian becomes

$$ \hat{H}_B = \frac{\hbar\omega}{2}(\hat{X}^2 + \hat{P}^2) \qquad (7.27) $$

$$ = \frac{\hbar\omega}{2}((\tilde{X} - \hat{A})^2 + \hat{P}^2) \qquad (7.28) $$

$$ = \frac{\hbar\omega}{2}(\tilde{X}^2 + \hat{A}^2 - 2\tilde{X}\hat{A} + \hat{P}^2) \qquad (7.29) $$

$$ \hat{H}_B = \frac{\hbar\omega}{2}(\tilde{X}^2 + \hat{P}^2) + \frac{\hbar\omega}{2}(\hat{A}^2 - 2\tilde{X}\hat{A}), \qquad (7.30) $$

and the interaction Hamiltonian becomes

$$ \hat{H}_{\text{int}} = g\hat{X}\hat{S} = g\tilde{X}\hat{S} - g\hat{A}\hat{S}. \qquad (7.31) $$

We choose $\hat{A} = \frac{g}{\hbar\omega}\hat{S}$ such that the new linear coupling term $g\tilde{X}\hat{S}$ is cancelled out by the linear term $-\hbar\omega\tilde{X}\hat{A}$ from the oscillator's Hamiltonian.

The Hamiltonian has now the following form

$$ \hat{H} = \hat{H}_S + \frac{\hbar\omega}{2}(\tilde{X}^2 + \hat{P}^2) + \frac{\hbar\omega}{2}\hat{A}^2 - g\hat{A}\hat{S} \qquad (7.32) $$

$$ = \hat{H}_S + \frac{\hbar\omega}{2}(\tilde{X}^2 + \hat{P}^2) + \frac{g^2}{2\hbar\omega}\hat{S}^2 - \frac{g^2}{\hbar\omega}\hat{S}^2 \qquad (7.33) $$

$$ \hat{H} = \hat{H}_S + \frac{\hbar\omega}{2}(\tilde{X}^2 + \hat{P}^2) - \frac{g^2}{2\hbar\omega}\hat{S}^2. \qquad (7.34) $$

We define the reorganisation energy $\lambda \stackrel{\text{def.}}{=} \frac{g^2}{2\hbar\omega}$. This reorganisation energy comes from a positive contribution from the oscillator's Hamiltonian and a negative contribution twice as large from the interaction Hamiltonian. Hence, at equilibrium $\langle\tilde{X}\rangle = 0$, the expectation value of the interaction Hamiltonian is $\langle\hat{H}_{\text{int}}\rangle = -2\lambda\langle S^2\rangle$.
> No. Never, not even when falling asleep, will I ever proudly mutter about nothing being able to surprise me. No. As one year has passed, so will another, and it will be just as rich in surprises as the first one... And so I have to go on dutifully learning.

Mikhail BULGAKOV,
A Young Doctor's Notebook

In this thesis, we have studied the behaviour of nanoscale quantum systems interacting with a macroscopic bosonic environment. Motivated by considerations from condensed matter physics and biological systems, we highlighted the need to go beyond the usual Markovian assumption and, furthermore, beyond the approximation (often taken for granted) of local independent baths. Indeed, miniaturised nanodevices have several working units in a small volume that thus dissipate energy in the same region of space. Similarly, biological 'nanochines', such as enzymes or pigment protein complexes, have different functional sites nanometers apart that hence dissipate energy in the very same environmental modes as well. In such cases, energy dissipated into the environment by a given sub-system can be absorbed later on by another sub-system – a highly non-Markovian process by nature. However, the description of such spatial correlations in the environment are known to be challenging. To tackle this problem, we have built upon a powerful numerically exact method named Time Evolving Density operator with Orthonormal Polynomials Algorithm (TEDOPA) that treats the system and the environment on an equal footing. By using tensor networks to represent the joint {System + Environment} quantum state as a Matrix Product State (MPS) this method is able to circumvent the curse of dimensionality inherent to many-body quantum systems.

In chapter 4, we introduced the chain mapping technique on which TEDOPA is based, and showed how it can be applied both at zero and finite temperature. From there, in order to describe spatial correlations, we have extended this technique to systems with an explicit spatial structure. The obtained chain mapping is what we call a correlated environment. By developing this new method, we made possible the description of multi-component quantum systems interacting with a
single common environment, and thus the study of spatiotemporal
correlations inside the environment.

On the methodology side of things, after developing our method to
handle a correlated environment, we studied in chapter 5 the influ-
ence of the ordering of bath modes in the MPS. This question stems
from the study of systems interacting with several distinct environ-
ments, for instance a hot and a cold bath. In these situations, it is
possible to organise the elements of the MPS in several ways. We
have shown that, for bosonic environments, the usual 'natural' way
to represent the MPS following the structure of the Hamiltonian is
not optimal, and we have introduced new ways of ordering the MPS
sites that are computationally more efficient. In doing so, we refuted a
common-sense assumption about the growth of MPS bond dimension
in multi-environment cases and made a conjecture on the origin of this
gain in efficiency.

Equipped with these new tools, we studied in chapter 6 the influence of
spatial correlations on the dynamics of a simple two-level system. We
have shown that such a system experiences non-Markovian revivals
of its initial state population that are associated with information
backflow from the environment. The environmental non-Markovian
signalling that we unveiled could shed light on biological processes,
for instance charge separation in reaction centres, where several mech-
anisms happening at the single excitation level are concerted over
different timescales.

Armed with our new understanding of environmental signalling, a
general model of dissipation in one-dimensional multi-component
systems has been studied in chapter 7. Looking at joint system/envi-
ronment observables, we have studied the out-of-equilibrium dynam-
ics induced by the excitation of a sub-system on another uncoupled
sub-system. Two different types of processes impacting the second
sub-system have been uncovered: (1) a transient energy harvesting
process enabling dynamically prohibited transitions, and (2) a per-
manent process where the reorganisation of the environment alters
the nature of the thermodynamically favoured ground state. In this
chapter, we also drew a parallel between the features of our general
dissipation model and the features of a biological regulatory process
called allostery.

The work presented in this thesis can lead to further developments
in several directions. The conjecture formulated in chapter 5 needs
further investigations to be either refuted or proven correct. In a similar
vein, more work is needed to understand the connection between the
ordering of the environmental modes when a system couples to several
environments and the growth of the state bond dimension. In the case
of several environments, it would also be of interest to discover how
non-additivity manifests itself in the chain dynamics.
The comparison of the exact results obtained with our numerical method with approximate treatments of the same problems would give us the opportunity to understand the open system’s dynamics from a different vantage point, and to highlight the essential features of the environmental dynamics resulting in the observed non-Markovian effects. This has partially been done by formulating a non-time-local master equation with a discrete memory kernel reflecting the spatial structure of the system in the environment response. Applying what we have called a ‘local Markov approximation’ leads to a time-delayed master equation able to recover the revivals observed in chapter 6.

Keeping a two qubit model such as the one presented in chapter 7, the range of new phenomena that can be investigated in a model as simple as this one has not been exhausted. Once driven, for example by an electromagnetic field, a TLS is an oscillator that oscillates at its autonomous Rabi frequency. Environmental signalling indirectly couples two oscillators that could thus synchronise to a new mutual frequency and phase lock, and possibly display other non-equilibrium phenomena. An experimental realisation of such a system could be done by encapsulating two quantum dots in a nanowire forming their common vibrational environment. Quantitative predictions could be made with the appropriate SD and compared with the dots emission spectra.

In the work presented here, biological systems were a motivation but no model of a specific biological system was studied. As we explained in chapter 1, there is a lot to gain in understanding how some crucial non-equilibrium biological processes are performed by a few correlated excitations. The electron bifurcation process, central to respiration, is an energy transduction mechanism where two electrons are spontaneously separated, the first one going down an energy gradient while the other one is climbing up another energy gradient. The electron gaining energy does so by reusing part of the energy dissipated by the first electron. Macroscopic thermodynamic models of this separation process have been proposed, but the microscopic mechanisms behind this out-of-equilibrium phenomenon are not yet understood. The tools we developed in this thesis are fit to describe such a system. It would require three local fermionic baths (one for the two electron donor and two for the electron acceptors at the end of each energy gradient) and a common vibrational bath. One of the difficulties of this project would be the parametrisation of the system Hamiltonian as the only accessible experimental data are the reduction potentials (or alternatively the reaction Gibbs free energy) of the constituents of the system.
<figure><img src="image_71.png" /></figure>
Bibliography

[1] Thibaut Lacroix et al. ‘Unveiling non-Markovian spacetime signaling in open quantum systems with long-range tensor network dynamics’. In: *Phys. Rev. A* 104.5 (Nov. 2021), p. 052204. DOI: 10.1103/PhysRevA.104.052204 (cited on pages xi, 6, 50, 75).

[2] Thibaut Lacroix, Brendon W. Lovett, and Alex W. Chin. *From Non-Markovian Dissipation to Spatiotemporal Control of Quantum Nanodevices*. Nov. 2022. DOI: 10.48550/arXiv.2205.11247 (cited on pages xi, 6, 91).

[3] Thibaut Lacroix. ‘Un modèle simple de décohérence’. In: *Bulletin de l’Union des Physiciens* (2023). Under Review (cited on pages xi, 6).

[4] Michael E. Cuffaro and Stephan Hartmann. *The Open Systems View*. arXiv:2112.11095 [physics, physics:quant-ph]. Dec. 2021 (cited on page 2).

[5] Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg. *Processus d'interaction entre photons et atomes*. Savoirs Actuels. EDP Sciences, Jan. 1996 (cited on page 2).

[6] Serge Haroche et al. *Exploring the Quantum: Atoms, Cavities, and Photons*. Oxford Graduate Texts. Oxford, New York: Oxford University Press, Aug. 2006 (cited on pages 2, 20).

[7] G. Lindblad. ‘On the generators of quantum dynamical semigroups’. In: *Commun.Math. Phys.* 48.2 (June 1976), pp. 119–130. DOI: 10.1007/BF01608499 (cited on page 2).

[8] Karl Blum. *Density Matrix Theory and Applications*. 3rd ed. Vol. 64. Springer Series on Atomic, Optical, and Plasma Physics. Springer, 2012 (cited on page 2).

[9] Ulrich Weiss. *Quantum Dissipative Systems*. en. 4th ed. World Scientific, Mar. 2012 (cited on pages 2, 14, 19, 64).

[10] A. O. Caldeira and A. J. Leggett. ‘Path integral approach to quantum Brownian motion’. In: *Physica A: Statistical Mechanics and its Applications* 121.3 (Sept. 1983), pp. 587–616. DOI: 10.1016/0378-4371(83)90013-4 (cited on page 2).

[11] A. J. Leggett et al. ‘Dynamics of the dissipative two-state system’. In: *Rev. Mod. Phys.* 59.1 (Jan. 1987), pp. 1–85. DOI: 10.1103/RevModPhys.59.1 (cited on page 2).

[12] R.P Feynman and F.L Vernon. ‘The theory of a general quantum system interacting with a linear dissipative system’. In: *Annals of Physics* 24 (Oct. 1963), pp. 118–173. DOI: 10.1016/0003-4916(63)90068-X (cited on page 2).

[13] Roman Orus. ‘A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States’. en. In: *Annals of Physics* 349 (Oct. 2014). arXiv: 1306.2164, pp. 117–158. DOI: 10.1016/j.aop.2014.06.013 (cited on pages 3, 27).

[14] Bob Coecke and Aleks Kissinger. *Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning*. ISBN: 9781107104228 Publisher: Cambridge University Press. Mar. 2017 (cited on pages 3, 26).

[15] Masoud Mohseni et al. *Quantum Effects in Biology*. Cambridge University Press, Aug. 2014 (cited on pages 3, 5, 50).

[16] Jonathan P. Dowling and Gerard J. Milburn. ‘Quantum technology: the second quantum revolution’. In: *Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences* 361.1809 (Aug. 2003). Ed. by A. G. J. MacFarlane, pp. 1655–1674. DOI: 10.1098/rsta.2003.1227 (cited on pages 3, 12, 91).

[17] Gregory S. Engel et al. ‘Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems’. en. In: *Nature* 446.7137 (Apr. 2007), pp. 782–786. DOI: 10.1038/nature05678 (cited on pages 3, 12, 75).
[18] G. Panitchayangkoon et al. 'Long-lived quantum coherence in photosynthetic complexes at physiologial temperature'. en. In: *Proceedings of the National Academy of Sciences* 107.29 (July 2010), pp. 12766-12770. DOI: 10.1073/pnas.1005484107 (cited on page 3).

[19] Elisabetta Collini et al. 'Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature'. en. In: *Nature* 463.7281 (Feb. 2010). Number: 7281 Publisher: Nature Publishing Group, pp. 644-647. DOI: 10.1038/nature08811 (cited on pages 4, 12, 75).

[20] Masoud Mohseni et al. 'Environment-assisted quantum walks in photosynthetic energy transfer'. en. In: *The Journal of Chemical Physics* 129.17 (Nov. 2008), p. 174106. DOI: 10.1063/1.3002335 (cited on pages 4, 12).

[21] M B Plenio and S F Huelga. 'Dephasing-assisted transport: quantum networks and biomolecules'. en. In: *New Journal of Physics* 10.11 (Nov. 2008), p. 113019. DOI: 10.1088/1367-2630/10/11/113019 (cited on page 4).

[22] F. Caruso et al. 'Highly efficient energy excitation transfer in light-harvesting complexes: The fundamental role of noise-assisted transport'. en. In: *The Journal of Chemical Physics* 131.10 (2009), p. 105106. DOI: 10.1063/1.3223548 (cited on page 4).

[23] A W Chin et al. 'Noise-assisted energy transfer in quantum networks and light-harvesting complexes'. en. In: *New Journal of Physics* 12.6 (June 2010), p. 065002. DOI: 10.1088/1367-2630/12/6/065002 (cited on pages 4, 45, 84).

[24] A. W. Chin, S. F. Huelga, and M. B. Plenio. 'Coherence and decoherence in biological systems: principles of noise-assisted transport and the origin of long-lived coherences'. en. In: *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences* 370.1972 (Aug. 2012), pp. 3638-3657. DOI: 10.1098/rsta.2011.0224 (cited on page 4).

[25] Elinor Zerah Harush and Yonatan Dubi. 'Do photosynthetic complexes use quantum coherence to increase their efficiency? Probably not'. In: *Science Advances* 7.8 (Feb. 2021). Publisher: American Association for the Advancement of Science, eabc4631. DOI: 10.1126/sciadv.abc4631 (cited on pages 4, 12).

[26] Nicole Yunger Halpern and David T. Limmer. 'Fundamental limitations on photoisomerization from thermodynamic resource theories'. In: *Phys. Rev. A* 101.4 (Apr. 2020). Publisher: American Physical Society, p. 042116. DOI: 10.1103/PhysRevA.101.042116 (cited on page 5).

[27] Giovanni Spaventa, Susana F. Huelga, and Martin B. Plenio. 'Capacity of non-Markovianity to boost the efficiency of molecular switches'. In: *Phys. Rev. A* 105.1 (Jan. 2022), p. 012420. DOI: 10.1103/PhysRevA.105.012420 (cited on page 5).

[28] Ivan H. Deutsch. 'Harnessing the Power of the Second Quantum Revolution'. In: *PRX Quantum* 1.2 (Nov. 2020), p. 020101. DOI: 10.1103/PRXQuantum.1.020101 (cited on pages 12, 91).

[29] Jianshu Cao et al. 'Quantum biology revisited'. In: *Science Advances* 6.14 (Apr. 2020). Publisher: American Association for the Advancement of Science, eaaz4888. DOI: 10.1126/sciadv.aaz4888 (cited on page 12).

[30] Stéphanie Valleau et al. 'Absence of Selection for Quantum Coherence in the Fenna–Matthews–Olson Complex: A Combined Evolutionary and Excitonic Study'. en. In: *ACS Cent. Sci.* 3.10 (Oct. 2017), pp. 1086–1095. DOI: 10.1021/acscentsci.7b00269 (cited on page 12).

[31] Jim Al-Khalili and Johnjoe McFadden. *Life on the Edge: The Coming of Age of Quantum Biology*. en. Google-Books-ID: u7D9oAEACAAJ. Bantam Press, 2014 (cited on page 12).

[32] Youngchan Kim et al. 'Quantum Biology: An Update and Perspective'. en. In: *Quantum Reports* 3.1 (Mar. 2021). Number: 1 Publisher: Multidisciplinary Digital Publishing Institute, pp. 80–126. DOI: 10.3390/quantum3010006 (cited on page 12).

[33] Neill Lambert et al. 'Quantum biology'. In: *Nature Phys* 9.1 (Jan. 2013), pp. 10–18. DOI: 10.1038/nphys2474 (cited on page 12).
[34] Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloë. *Mécanique Quantique - Tome 1*. 2nd ed. Vol. 1. Savoirs Actuels. EDP Sciences (cited on page 13).

[35] Heinz-Peter Breuer and Francesco Petruccione. *The Theory of Open Quantum Systems*. en_US. Oxford University Press, Jan. 2007 (cited on pages 14, 19, 64, 81, 97).

[36] Gernot Schaller. *Open Quantum Systems Far from Equilibrium*. Vol. 881. Lecture Notes in Physics. Cham: Springer International Publishing, 2014 (cited on page 14).

[37] Walter Appel. *Mathematics for Physics and Physicists*. en. Princeton University Press, Mar. 2007 (cited on pages 16, 44, 45).

[38] Li Li, Michael J. W. Hall, and Howard M. Wiseman. 'Concepts of quantum non-Markovianity: A hierarchy'. en. In: *Physics Reports*. Concepts of quantum non-Markovianity: A hierarchy 759 (Oct. 2018), pp. 1-51. DOI: 10.1016/j.physrep.2018.07.001 (cited on page 20).

[39] Florian A. Y. N. Schröder et al. 'Tensor network simulation of multi-environmental open quantum dynamics via machine learning and entanglement renormalisation'. en. In: *Nat Commun* 10.1 (Mar. 2019). Number: 1 Publisher: Nature Publishing Group, p. 1062. DOI: 10.1038/s41467-019-09039-7 (cited on pages 20, 93, 107).

[40] Graham A. Worth and Lorenz S. Cederbaum. 'Beyond Born-Oppenheimer: Molecular Dynamics Through a Conical Intersection'. en. In: *Annu. Rev. Phys. Chem.* 55.1 (June 2004), pp. 127-158. DOI: 10.1146/annurev.physchem.55.091602.094335 (cited on pages 20, 93).

[41] David M. Leitner. 'Energy Flow in Proteins'. en. In: *Annu. Rev. Phys. Chem.* 59.1 (May 2008), pp. 233-259. DOI: 10.1146/annurev.physchem.59.032607.093606 (cited on pages 20, 93).

[42] Philipp Strasberg et al. 'Nonequilibrium thermodynamics in the strong coupling and non-Markovian regime based on a reaction coordinate mapping'. en. In: *New J. Phys.* 18.7 (July 2016), p. 073007. DOI: 10.1088/1367-2630/18/7/073007 (cited on page 21).

[43] Neill Lambert et al. 'Modelling the ultra-strongly coupled spin-boson model with unphysical modes'. en. In: *Nat Commun* 10.1 (Dec. 2019), p. 3721. DOI: 10.1038/s41467-019-11656-1 (cited on page 21).

[44] Yoshitaka Tanimura. 'Numerically "exact" approach to open quantum dynamics: The hierarchical equations of motion (HEOM)'. In: *J. Chem. Phys.* 153.2 (July 2020). Publisher: American Institute of Physics, p. 020901. DOI: 10.1063/5.0011599 (cited on page 21).

[45] Dmitrii E. Makarov and Nancy Makri. 'Path integrals for dissipative systems by tensor multiplication. Condensed phase quantum dynamics for arbitrarily long time'. In: *Chemical Physics Letters* 221.5 (Apr. 1994), pp. 482-491. DOI: 10.1016/0009-2614(94)00275-4 (cited on page 21).

[46] A. Strathearn et al. 'Efficient non-Markovian quantum dynamics using time-evolving matrix product operators'. en. In: *Nature Communications* 9.1 (Dec. 2018), p. 3322. DOI: 10.1038/s41467-018-05617-3 (cited on pages 21, 48, 81).

[47] Moritz Cygorek et al. 'Simulation of open quantum systems by automated compression of arbitrary environments'. In: *Nat. Phys.* (Mar. 2022). Publisher: Nature Publishing Group, pp. 1-7. DOI: 10.1038/s41567-022-01544-9 (cited on pages 21, 48).

[48] Javier Prior et al. 'Efficient Simulation of Strong System-Environment Interactions'. en. In: *Phys. Rev. Lett.* 105.5 (July 2010), p. 050404. DOI: 10.1103/PhysRevLett.105.050404 (cited on pages 21, 43, 47).

[49] Roger Penrose. 'Applications of negative dimensional tensors'. In: *Combinatorial mathematics and its applications* 1 (1971), pp. 221-224 (cited on page 26).

[50] Steven R. White. 'Density matrix formulation for quantum renormalization groups'. In: *Phys. Rev. Lett.* 69.19 (Nov. 1992), pp. 2863-2866. DOI: 10.1103/PhysRevLett.69.2863 (cited on page 26).

[51] Ulrich Schollwöck. 'The density-matrix renormalization group'. In: *Rev. Mod. Phys.* 77.1 (Apr. 2005). arXiv: cond-mat/0409292, pp. 259-315. DOI: 10.1103/RevModPhys.77.259 (cited on page 26).
[52] Ulrich Schollwöck. 'The density-matrix renormalization group in the age of matrix product states'. In: *Annals of Physics* 326.1 (Jan. 2011), pp. 96–192. DOI: 10.1016/j.aop.2010.09.012 (cited on page 26).

[53] F. Verstraete and J. I. Cirac. *Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions*. arXiv:cond-mat/0407066. July 2004 (cited on pages 26, 37).

[54] G. Vidal. 'Class of Quantum Many-Body States That Can Be Efficiently Simulated'. In: *Phys. Rev. Lett.* 101.11 (Sept. 2008). Publisher: American Physical Society, p. 110501. DOI: 10.1103/PhysRevLett.101.110501 (cited on page 26).

[55] L. Tagliacozzo, G. Evenbly, and G. Vidal. 'Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law'. In: *Phys. Rev. B* 80.23 (Dec. 2009), p. 235127. DOI: 10.1103/PhysRevB.80.235127 (cited on page 26).

[56] G. Evenbly and G. Vidal. 'Tensor network states and geometry'. In: *J Stat Phys* 145.4 (Nov. 2011). arXiv:1106.1082, pp. 891–918. DOI: 10.1007/s10955-011-0237-4 (cited on page 26).

[57] Guifré Vidal. 'Efficient Simulation of One-Dimensional Quantum Many-Body Systems'. In: *Phys. Rev. Lett.* 93.4 (July 2004). Publisher: American Physical Society, p. 040502. DOI: 10.1103/PhysRevLett.93.040502 (cited on pages 26, 37).

[58] Jutho Haegeman et al. 'Time-Dependent Variational Principle for Quantum Lattices'. In: *Phys. Rev. Lett.* 107.7 (Aug. 2011), p. 070601. DOI: 10.1103/PhysRevLett.107.070601 (cited on pages 26, 38).

[59] Michael P. Zaletel et al. 'Time-evolving a matrix product state with long-ranged interactions'. In: *Phys. Rev. B* 91.16 (Apr. 2015). Publisher: American Physical Society, p. 165112. DOI: 10.1103/PhysRevB.91.165112 (cited on pages 26, 37).

[60] Sebastian Paeckel et al. 'Time-evolution methods for matrix-product states'. en. In: *Annals of Physics* 411 (Dec. 2019), p. 167998. DOI: 10.1016/j.aop.2019.167998 (cited on pages 26, 34, 37, 38, 40, 68).

[61] David Poulin et al. 'Quantum Simulation of Time-Dependent Hamiltonians and the Convenient Illusion of Hilbert Space'. In: *Phys. Rev. Lett.* 106.17 (Apr. 2011), p. 170501. DOI: 10.1103/PhysRevLett.106.170501 (cited on page 27).

[62] Mark Srednicki. 'Entropy and area'. In: *Phys. Rev. Lett.* 71.5 (Aug. 1993), pp. 666–669. DOI: 10.1103/PhysRevLett.71.666 (cited on page 27).

[63] G. Vidal et al. 'Entanglement in Quantum Critical Phenomena'. In: *Phys. Rev. Lett.* 90.22 (June 2003), p. 227902. DOI: 10.1103/PhysRevLett.90.227902 (cited on page 27).

[64] Michael M. Wolf et al. 'Area Laws in Quantum Systems: Mutual Information and Correlations'. In: *Phys. Rev. Lett.* 100.7 (Feb. 2008), p. 070502. DOI: 10.1103/PhysRevLett.100.070502 (cited on page 27).

[65] Frank Verstraete, Michael M. Wolf, and J. Ignacio Cirac. 'Quantum computation and quantum-state engineering driven by dissipation'. In: *Nature Phys* 5.9 (Sept. 2009), pp. 633–636. DOI: 10.1038/nphys1342 (cited on pages 37, 92).

[66] Masuo Suzuki. 'Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems'. In: *Commun. Math. Phys.* 51 (June 1976), pp. 183–190. DOI: 10.1007/BF01609348 (cited on page 37).

[67] Yousef Saad. *Iterative Methods for Sparse Linear Systems*. Other Titles in Applied Mathematics. Society for Industrial and Applied Mathematics, Jan. 2003 (cited on page 38).

[68] P. A. M. Dirac. 'Note on Exchange Phenomena in the Thomas Atom'. In: *Mathematical Proceedings of the Cambridge Philosophical Society* 26.3 (July 1930). Publisher: Cambridge University Press, pp. 376–385. DOI: 10.1017/S0305004100016108 (cited on page 38).

[69] Yakov Frenkel. *Wave mechanics, advanced general theory*. Vol. 1. Oxford, 1934 (cited on page 38).

[70] Jutho Haegeman et al. 'Unifying time evolution and optimization with matrix product states'. In: *Phys. Rev. B* 94.16 (Oct. 2016), p. 165116. DOI: 10.1103/PhysRevB.94.165116 (cited on pages 38, 39).
[71] A. Raab. 'On the Dirac-Frenkel/McLachlan variational principle'. In: *Chemical Physics Letters* 319.5 (Mar. 2000), pp. 674-678. DOI: 10.1016/S0009-2614(00)00200-1 (cited on page 38).

[72] Angus J. Dunnett and Alex W. Chin. 'Efficient bond-adaptive approach for finite-temperature open quantum dynamics using the one-site time-dependent variational principle for matrix product states'. In: *Phys. Rev. B* 104.21 (Dec. 2021), p. 214302. DOI: 10.1103/PhysRevB.104.214302 (cited on pages 40, 61, 62, 68).

[73] Alex W. Chin et al. 'Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials'. en. In: *Journal of Mathematical Physics* 51.9 (Sept. 2010). arXiv: 1006.4507, p. 092109. DOI: 10.1063/1.3490188 (cited on pages 43, 45).

[74] M. P. Woods, M. Cramer, and M. B. Plenio. 'Simulating Bosonic Baths with Error Bars'. In: *Phys. Rev. Lett.* 115.13 (Sept. 2015), p. 130401. DOI: 10.1103/PhysRevLett.115.130401 (cited on pages 43, 47).

[75] Walter Gautschi. 'Algorithm 726: ORTHPOL-a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules'. In: *ACM Trans. Math. Softw.* 20.1 (Mar. 1994), pp. 21-62. DOI: 10.1145/174603.174605 (cited on page 45).

[76] Angus Dunnett and Thibaut Lacroix. *MPSDynamics*. July 2021. DOI: 10.5281/zenodo.5106435. (Visited on 01/18/2022) (cited on pages 45, 69).

[77] F. Verstraete, J. J. García-Ripoll, and J. I. Cirac. 'Matrix Product Density Operators: Simulation of Finite-Temperature and Dissipative Systems'. In: *Phys. Rev. Lett.* 93.20 (Nov. 2004), p. 207204. DOI: 10.1103/PhysRevLett.93.207204 (cited on page 47).

[78] D. Tamascelli et al. 'Efficient Simulation of Finite-Temperature Open Quantum Systems'. In: *Phys. Rev. Lett.* 123.9 (Aug. 2019), p. 090402. DOI: 10.1103/PhysRevLett.123.090402 (cited on pages 48, 49, 58, 86).

[79] Angus J. Dunnett and Alex W. Chin. 'Simulating Quantum Vibronic Dynamics at Finite Temperatures With Many Body Wave Functions at 0 K'. In: *Front. Chem.* 8 (2021). Publisher: Frontiers. DOI: 10.3389/fchem.2020.600731 (cited on page 49).

[80] Yasushi Takahashi and Hiroomi Umezawa. 'Thermo field dynamics'. In: *Int. J. Mod. Phys. B* 10.13n14 (June 1996). Publisher: World Scientific Publishing Co., pp. 1755-1805. DOI: 10.1142/S0217979296000817 (cited on page 56).

[81] Maxim F. Gelin and Raffaele Borrelli. 'Thermal Schrödinger Equation: Efficient Tool for Simulation of Many-Body Quantum Dynamics at Finite Temperature'. In: *Annalen der Physik* 529.12 (2017), p. 1700200. DOI: 10.1002/andp.201700200 (cited on page 56).

[82] Florian A. Y. N. Schröder et al. 'Tensor network simulation of multi-environmental open quantum dynamics via machine learning and entanglement renormalisation'. In: *Nat Commun* 10.1 (Mar. 2019). Number: 1 Publisher: Nature Publishing Group, p. 1062. DOI: 10.1038/s41467-019-09039-7 (cited on page 59).

[83] Angus J. Dunnett et al. 'Influence of non-adiabatic effects on linear absorption spectra in the condensed phase: Methylene blue'. In: *J. Chem. Phys.* 155 (Oct. 2021). arXiv:2107.06587 [physics, physics:quant-ph], p. 144112. DOI: 10.1063/5.0062950 (cited on pages 59, 107).

[84] G. Vidal. 'Entanglement Renormalization'. In: *Phys. Rev. Lett.* 99.22 (Nov. 2007), p. 220405. DOI: 10.1103/PhysRevLett.99.220405 (cited on page 59).

[85] Weitang Li et al. 'On the fly swapping algorithm for ordering of degrees of freedom in density matrix renormalization group'. In: *J. Phys.: Condens. Matter* 34.25 (June 2022), p. 254003. DOI: 10.1088/1361-648X/ac640e (cited on page 60).

[86] Lucas Kohn and Giuseppe E. Santoro. 'Efficient mapping for Anderson impurity problems with matrix product states'. In: *Phys. Rev. B* 104.1 (July 2021), p. 014303. DOI: 10.1103/PhysRevB.104.014303 (cited on pages 60, 62).
[87] Lucas Kohn and Giuseppe E Santoro. 'Quench dynamics of the Anderson impurity model at finite temperature using matrix product states: entanglement and bath dynamics'. In: *J. Stat. Mech.* 2022.6 (June 2022), p. 063102. DOI: 10.1088/1742-5468/ac729b (cited on pages 60, 61).

[88] Gerald E. Fux et al. *Thermalization of a spin chain strongly coupled to its environment*. arXiv:2201.05529 [cond-mat, physics:quant-ph]. Jan. 2022. DOI: 10.48550/arXiv.2201.05529 (cited on page 61).

[89] Dominic Gribben et al. 'Exact dynamics of non-additive environments in non-Markovian open quantum systems'. In: *PRX Quantum* 3.1 (Feb. 2022). arXiv:2109.08442 [quant-ph], p. 010321. DOI: 10.1103/PRXQuantum.3.010321 (cited on page 61).

[90] A. W. Chin et al. 'The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigment-protein complexes'. In: *Nature Phys* 9.2 (Feb. 2013). Number: 2 Publisher: Nature Publishing Group, pp. 113-118. DOI: 10.1038/nphys2515 (cited on page 75).

[91] Christoph Kreisbeck and Tobias Kramer. 'Long-Lived Electronic Coherence in Dissipative Exciton Dynamics of Light-Harvesting Complexes'. In: *J. Phys. Chem. Lett.* 3.19 (Oct. 2012). Publisher: American Chemical Society, pp. 2828-2833. DOI: 10.1021/jz3012029 (cited on page 75).

[92] Frank Müh and Athina Zouni. 'The nonheme iron in photosystem II'. In: *Photosynth Res* 116.2-3 (Oct. 2013), pp. 295-314. DOI: 10.1007/s11120-013-9926-y (cited on pages 76, 92, 93).

[93] Olga Bozovic et al. 'Real-time observation of ligand-induced allosteric transitions in a PDZ domain'. In: *Proceedings of the National Academy of Sciences* 117.42 (Oct. 2020). Publisher: Proceedings of the National Academy of Sciences, pp. 26031-26039. DOI: 10.1073/pnas.2012999117 (cited on page 76).

[94] Jingjing Guo and Huan-Xiang Zhou. 'Protein Allostery and Conformational Dynamics'. In: *Chem. Rev.* 116.11 (June 2016). Publisher: American Chemical Society, pp. 6503-6515. DOI: 10.1021/acs.chemrev.5b00590 (cited on page 76).

[95] Harold Ollivier, David Poulin, and Wojciech H. Zurek. 'Environment as a witness: Selective proliferation of information and emergence of objectivity in a quantum universe'. In: *Phys. Rev. A* 72.4 (Oct. 2005), p. 042113. DOI: 10.1103/PhysRevA.72.042113 (cited on page 85).

[96] Dario Tamascelli. 'Excitation Dynamics in Chain-Mapped Environments'. In: *Entropy* 22.11 (Nov. 2020). Number: 11 Publisher: Multidisciplinary Digital Publishing Institute, p. 1320. DOI: 10.3390/ e22111320 (cited on page 87).

[97] *Quantum Computation and Quantum Information: 10th Anniversary Edition*. ISBN: 9780511976667 Publisher: Cambridge University Press. Dec. 2010 (cited on page 91).

[98] Pascal Degiovanni et al. *Physique quantique, information et calcul - Des concepts aux applications*. 1st. Savoirs Actuels. EDP Sciences, Jan. 2020 (cited on page 91).

[99] Masahito Hayashi. *Quantum Information*. 1st ed. Springer Berlin Heidelberg, 2006 (cited on page 91).

[100] Gilbert Grynberg, Alain Aspect, and Claude Fabre. *Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light*. Cambridge: Cambridge University Press, 2010 (cited on page 91).

[101] Pieter Kok and Brendon W. Lovett. *Introduction to Optical Quantum Information Processing*. Cambridge: Cambridge University Press, 2010 (cited on page 91).

[102] Markus Aspelmeyer, Tobias J. Kippenberg, and Florian Marquardt, eds. *Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light*. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014 (cited on page 91).

[103] Hamidreza Esmaielpour et al. 'Hot carrier relaxation and inhibited thermalization in superlattice heterostructures: The potential for phonon management'. In: *Applied Physics Letters* 118.21 (2021), p. 213902 (cited on page 91).

[104] Masoud Mohseni et al. 'Environment-assisted quantum walks in photosynthetic energy transfer'. In: *The Journal of Chemical Physics* 129.17 (Nov. 2008), p. 174106. DOI: 10.1063/1.3002335 (cited on page 92).
[105] M B Plenio and S F Huelga. 'Dephasing-assisted transport: quantum networks and biomolecules'. In: *New J. Phys.* 10.11 (Nov. 2008), p. 113019. DOI: 10.1088/1367-2630/10/11/113019 (cited on page 92).

[106] F. Caruso et al. 'Highly efficient energy excitation transfer in light-harvesting complexes: The fundamental role of noise-assisted transport'. In: *J. Chem. Phys.* 131.10 (2009), p. 105106. DOI: 10.1063/1.3223548 (cited on page 92).

[107] Melina Wertnik et al. 'Optimizing co-operative multi-environment dynamics in a dark-state-enhanced photosynthetic heat engine'. In: *The Journal of chemical physics* 149.8 (2018), p. 084112 (cited on page 92).

[108] Srijon Ghosh et al. 'Fast charging of a quantum battery assisted by noise'. In: *Physical Review A* 104.3 (2021), p. 032207 (cited on page 92).

[109] Anton Potočnik et al. 'Studying light-harvesting models with superconducting circuits'. In: *Nature communications* 9.1 (2018), pp. 1-7 (cited on page 92).

[110] Christine Maier et al. 'Environment-assisted quantum transport in a 10-qubit network'. In: *Physical Review Letters* 122.5 (2019), p. 050501 (cited on page 92).

[111] Jack Hansom et al. 'Environment-assisted quantum control of a solid-state spin via coherent dark states'. In: *Nature Physics* 10.10 (2014), pp. 725-730 (cited on page 92).

[112] Ronnie Kosloff. 'Quantum thermodynamics and open-systems modeling'. In: *The Journal of chemical physics* 150.20 (2019), p. 204105 (cited on page 92).

[113] Sebastian Deffner and Steve Campbell. *Quantum Thermodynamics*. Morgan & Claypool, July 2019 (cited on pages 92, 97).

[114] A. Bermudez, T. Schaetz, and M. B. Plenio. 'Dissipation-Assisted Quantum Information Processing with Trapped Ions'. In: *Phys. Rev. Lett.* 110.11 (Mar. 2013), p. 110502. DOI: 10.1103/PhysRevLett.110.110502 (cited on page 92).

[115] S. Gröblacher et al. 'Observation of non-Markovian micromechanical Brownian motion'. In: *Nat Commun* 6.1 (Nov. 2015), p. 7606. DOI: 10.1038/ncomms8606 (cited on page 92).

[116] C-F Li, G-C Guo, and J Piilo. 'Non-Markovian quantum dynamics: What is it good for?' In: *EPL (Europhysics Letters)* 128.3 (2020), p. 30001 (cited on page 92).

[117] Bi-Heng Liu et al. 'Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems'. In: *Nature Physics* 7.12 (2011), pp. 931-934 (cited on page 92).

[118] Deepak Khurana, Bijay Kumar Agarwalla, and TS Mahesh. 'Experimental emulation of quantum non-Markovian dynamics and coherence protection in the presence of information backflow'. In: *Physical Review A* 99.2 (2019), p. 022107 (cited on page 92).

[119] Kristian Høeg Madsen et al. 'Observation of non-Markovian dynamics of a single quantum dot in a micropillar cavity'. In: *Physical review letters* 106.23 (2011), p. 233601 (cited on page 92).

[120] Mohan Sarovar et al. 'Detecting crosstalk errors in quantum information processors'. In: *Quantum* 4 (Sept. 2020). arXiv:1908.09855 [quant-ph], p. 321. DOI: 10.22331/q-2020-09-11-321 (cited on page 92).

[121] Raj Pandya et al. 'Optical Projection and Spatial Separation of Spin-Entangled Triplet Pairs from the S1 (21 Ag-) State of Pi-Conjugated Systems'. In: *Chem* 6.10 (2020), pp. 2826-2851 (cited on page 93).

[122] Ángel Rivas, Susana F Huelga, and Martin B Plenio. 'Quantum non-Markovianity: characterization, quantification and detection'. In: *Rep. Prog. Phys.* 77.9 (Sept. 2014), p. 094001. DOI: 10.1088/0034-4885/77/9/094001 (cited on page 92).

[123] Inés De Vega and Daniel Alonso. 'Dynamics of non-Markovian open quantum systems'. In: *Reviews of Modern Physics* 89.1 (2017), p. 015001 (cited on page 92).
[124] Jonathon L. Yuly, Peng Zhang, and David N. Beratan. 'Energy transduction by reversible electron bifurcation'. In: *Current Opinion in Electrochemistry* 29 (Oct. 2021), p. 100767. DOI: 10.1016/j.coelec.2021.100767 (cited on pages 92, 107).

[125] Marten L Chaillet et al. 'Static Disorder in Excitation Energies of the Fenna–Matthews–Olson Protein: Structure-Based Theory Meets Experiment'. In: *J. Phys. Chem. Lett.* 11.24 (2020), pp. 10306–10314 (cited on page 92).

[126] Vincent Fourmond et al. 'Understanding and Design of Bidirectional and Reversible Catalysts of Multielectron, Multistep Reactions'. In: *J. Am. Chem. Soc.* 141.28 (July 2019). Publisher: American Chemical Society, pp. 11269–11285. DOI: 10.1021/jacs.9b04854 (cited on page 92).

[127] Miloš Đokić and Han Sen Soo. 'Artificial photosynthesis by light absorption, charge separation, and multielectron catalysis'. In: *Chem. Commun.* 54.50 (June 2018). Publisher: The Royal Society of Chemistry, pp. 6554–6572. DOI: 10.1039/C8CC02156B (cited on page 92).

[128] Rui Wang et al. 'Gate tunable hole charge qubit formed in a Ge/Si nanowire double quantum dot coupled to microwave photons'. In: *Nano Letters* 19.2 (2019), pp. 1052–1060 (cited on page 93).

[129] O. Arcizet et al. 'A single nitrogen-vacancy defect coupled to a nanomechanical oscillator'. In: *Nature Phys* 7.11 (Nov. 2011), pp. 879–883. DOI: 10.1038/nphys2070 (cited on page 94).

[130] I. Yeo et al. 'Strain-mediated coupling in a quantum dot–mechanical oscillator hybrid system'. In: *Nature Nanotech* 9.2 (Feb. 2014), pp. 106–110. DOI: 10.1038/nnano.2013.274 (cited on page 94).

[131] Philipp Treutlein et al. 'Hybrid Mechanical Systems'. In: *Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light*. Ed. by Markus Aspelmeyer, Tobias J. Kippenberg, and Florian Marquardt. Quantum Science and Technology. Berlin, Heidelberg: Springer, 2014, pp. 327–351. DOI: 10.1007/978-3-642-55312-7_14 (cited on page 94).

[132] Anna Köhler and Bässler Heinz. *Electronic Processes in Organic Semiconductors: An Introduction*. Wiley, Mar. 2015 (cited on page 94).

[133] Jacques Monod, Jean-Pierre Changeux, and François Jacob. 'Allosteric proteins and cellular control systems'. In: *Journal of Molecular Biology* 6.4 (Apr. 1963), pp. 306–329. DOI: 10.1016/S0022-2836(63)80091-1 (cited on page 105).

[134] Jean-Pierre Changeux. '50th anniversary of the word “allosteric”'. In: *Protein Science* 20.7 (2011). _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/pro.658, pp. 1119–1124. DOI: 10.1002/pro.658 (cited on pages 105, 107).

[135] Jin Liu and Ruth Nussinov. 'Allostery: An Overview of Its History, Concepts, Methods, and Applications'. In: *PLoS Comput Biol* 12.6 (June 2016). DOI: 10.1371/journal.pcbi.1004966 (cited on pages 105, 107).

[136] Chung-Jung Tsai and Ruth Nussinov. 'A Unified View of “How Allostery Works”'. In: *PLoS Comput Biol* 10.2 (Feb. 2014). Ed. by Dennis R. Livesay, e1003394. DOI: 10.1371/journal.pcbi.1003394 (cited on page 105).

[137] David M. Leitner, Changbong Hyeon, and Korey M. Reid. 'Water-mediated biomolecular dynamics and allostery'. In: *J. Chem. Phys.* 152.24 (June 2020). Publisher: American Institute of Physics, p. 240901. DOI: 10.1063/5.0011392 (cited on page 105).

[138] Holger Flechsig. 'Design of Elastic Networks with Evolutionary Optimized Long-Range Communication as Mechanical Models of Allosteric Proteins'. In: *Biophysical Journal* 113.3 (Aug. 2017), pp. 558–571. DOI: 10.1016/j.bpj.2017.06.043 (cited on page 105).

[139] Jason W. Rocks et al. 'Designing allostery-inspired response in mechanical networks'. In: *Proceedings of the National Academy of Sciences* 114.10 (Feb. 2017), pp. 2520–2525. DOI: 10.1073/pnas.1612139114. (Visited on 05/10/2022) (cited on page 105).

[140] Samuel Hertig, Naomi R. Latorraca, and Ron O. Dror. 'Revealing Atomic-Level Mechanisms of Protein Allostery with Molecular Dynamics Simulations'. In: *PLoS Comput Biol* 12.6 (June 2016). Ed. by Jin Liu, e1004746. DOI: 10.1371/journal.pcbi.1004746 (cited on page 105).
[141] Hesam N. Motlagh et al. 'The ensemble nature of allostery'. en. In: *Nature* 508.7496 (Apr. 2014), pp. 331–339. DOI: 10.1038/nature13001 (cited on page 105).

[142] Vincent J. Hilser, James O. Wrabl, and Hesam N. Motlagh. 'Structural and Energetic Basis of Allostery'. In: *Annu. Rev. Biophys.* 41.1 (June 2012), pp. 585–609. DOI: 10.1146/annurev-biophys-050511-102319 (cited on pages 105, 107).

[143] Shoshana J. Wodak et al. 'Allostery in Its Many Disguises: From Theory to Applications'. In: *Structure* 27.4 (2019), pp. 566–578. DOI: 10.1016/j.str.2019.01.003 (cited on page 105).

[144] Jacques Monod. *Le Hasard et la Nécessité. Essai sur la philosophie naturelle de la biologie moderne. Sciences humaines*. Seuil, Oct. 1970 (cited on page 105).

[145] Giulio Chiribella et al. 'Quantum computations without definite causal structure'. In: *Phys. Rev. A* 88 (2 Aug. 2013), p. 022318. DOI: 10.1103/PhysRevA.88.022318 (cited on page 107).

[146] Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. 'Quantum correlations with no causal order'. In: *Nat Commun* 3.1 (Oct. 2012). Number: 1 Publisher: Nature Publishing Group, p. 1092. DOI: 10.1038/ncomms2076 (cited on page 107).

[147] Thomas Renger et al. 'Normal mode analysis of the spectral density of the Fenna–Matthews–Olson light-harvesting protein: how the protein dissipates the excess energy of excitons'. In: *J. Phys. Chem. B* 116.50 (2012), pp. 14565–14580 (cited on page 107).

[148] Sarah E Morgan, Daniel J Cole, and Alex W Chin. 'Nonlinear network model analysis of vibrational energy transfer and localisation in the Fenna-Matthews-Olson complex'. In: *Sci. Rep.* 6.1 (2016), pp. 1–10 (cited on page 107).

[149] David M. Leitner. 'Vibrational Energy Transfer in Helices'. In: *Phys. Rev. Lett.* 87.18 (Oct. 2001), p. 188102. DOI: 10.1103/PhysRevLett.87.188102 (cited on page 107).