*3.3. Open-Quantum Systems*

The theory of open-quantum systems (OQS) is a well-established model of the coupling between a system and an external environment [78–80]. Usually, the latter is considered to be a large reservoir of particles, momentum, and energy which it freely exchanges with the system. The coupling between the

system and the environment might be quite general, and we can couple more than one environment, with different macroscopic thermodynamic parameters, such as temperature, chemical potential, or pressure, at a time. Usually the theory is used to describe the dynamical relaxation of the system towards some steady state or thermal equilibrium, but it has found also widespread applications in quantum optics, transport modeling, surface hopping in chemical reactions [81], and so forth (see also Section 3.4).

In the following, we are making some standard assumptions about the environment(s):


An efficient way to deal with the environment is therefore by tracing out its degrees of freedom and introduce effective correlation functions, (usually dependent on the thermodynamic parameters) with which we describe the dynamics of the quantum system. The way the tracing is made defines the accuracy of the theory. Normally, one can identify two large families of approximations: in the first (called Markovian) the system dynamics does not have any history. The state is determined solely by the evolution at a certain time *t* and does not depend on any time *t* < *t*. On the other hand, we have non-Markovian dynamics where the state of the system depends on all or some of the previous times, the history of the system, since the initial time *t*0. Into this second family, we then distinguish how the kernel generated by the coupling with the environment depends on the actual state of the system. Let us therefore establish some notation and the standard results. In the following, we will work on a formulation based on the density matrix. Alternative formulations based on a vector state in the Hilbert space are possible and will be briefly discussed later [78,80].

We assume that a system *S* is coupled to an environment *B*. When there is no coupling, the dynamics of *S* is described by a set of operators acting on a Hilbert space H*S*, while the dynamics of *B* by operators acting on H*<sup>B</sup>*. We further assume that these Hilbert spaces are disjoint, therefore the total Hilbert space of *S* + *B* is given by H*S* ⊗ H*<sup>B</sup>*, and thus the state of the total system *S* + *B* is represented by a density matrix in this space. It follows that each operator of *S* commutes with each operator of *B*. We now assume that there is a coupling between *S* and *B*, *HSB* = ∑*qi λiVi*,*<sup>S</sup>* ⊗ *Vi*,*<sup>B</sup>* where *Vi*,*<sup>S</sup>*(*B*) is an operator acting on the <sup>H</sup>*S*(*B*) space, and *q* the number of operators coupling the system with the bath. Notice that this expansion is always possible due to the commutativity between the operators acting on *S* and *B*. For the moment, we focus on the simple case *q* = 1, but the extension to the more general case *q* > 1 is trivial, so in the following we set *λi* = *λ*. The density matrix of the total system evolves according to the von-Neumann equation,

$$
\partial\_l \rho(t) = -\frac{i}{\hbar} \left[ H, \rho(t) \right], \tag{47}
$$

where *H* = *HS* + *HB* + *HSB* is the total Hamiltonian. A basis set for the total Hilbert space is given by {|*j*, *k*} where *j* is a element of the basis for H*S* and *k* for H*<sup>B</sup>*.

Clearly, if we are interested in the dynamics of the system *S* only, this equation contains much more information than needed, since it entails the dynamics of the many degrees of freedom of the environment. We wish therefore to obtain an equation of motion for a density matrix, where the coupling with the environment is effectively described by some macroscopic parameters. This is possible when one assumes that the coupling strength *λ* is sufficiently small and we can use

perturbation theory. The exact definition of the "smallness" of *λ* is actually an open problem, and we will refer the interested reader elsewhere [78–80].

The aim of the theory is to obtain the dynamics of the reduced density matrix, whose element (*i*, *j*) is given by

$$(\rho\_R)\_{i,j} = (\text{tr}\_{\mathbb{B}}\rho)\_{i,j} = \sum\_{k \in \mathcal{H}\_B} \langle i,k|\rho|k,j\rangle. \tag{48}$$

The reduced density matrix is therefore a density matrix in the space of the system *S*, but its dynamics is determined not only by *HS* but also by the dynamics of the environment. The derivation of the equation of motion for the reduced density matrix of the system stems from the dynamics of the density matrix of the bath when decoupled from the system, *ρB*(*t*) = exp (−*iHBt*) *ρB*(0) exp (*iHBt*), the factorization of the initial density matrix *ρ*(0) = *ρS*(0) ⊗ *ρB*(0), and the vanishing of the quantum averages of the bath operators *VB* to first order in *λ*, tr*B* (*ρBVB*) ∝ *λ*<sup>2</sup> (this latter assumption can be relaxed and would eventually contribute an effective force that redefines the system Hamiltonian). This standard procedure leads, after some straightforward algebra, to [78,80]

$$\frac{d\rho\_S}{dt} = -i\left[H\_{S'}\rho\_S\right] + \lambda^2 \left[V\_{S'}M^\dagger(t) - M(t)\right],\tag{49}$$

where we have defined

$$M(t) = \int\_0^t dt' \mathcal{C}(t, t') e^{-iH\_\mathcal{S}(t - t')} V\_\mathcal{S} \rho\_\mathcal{S}(t') e^{iH\_\mathcal{S}(t - t')} \tag{50}$$

and

$$\mathbb{C}(t, t') = \text{tr}\_B \left[ \rho\_B(0) V\_B(t) V\_B(t') \right] \tag{51}$$

is the bath correlation function, where *VB*(*t*) = exp(*iHBt*)*VB*(0) exp(−*iHBt*) is the time evolution of the bath operators. Equation (49) is difficult to solve for two reasons. On the one hand, the density matrix is usually a dense matrix of *N*<sup>2</sup> elements, if we have *N* elements in the basis set. This usually requires long computation time and large amounts of memory. On the other hand, it contains the full history of the system and at each time step this history needs to be evaluated to calculate the integrals in Equation (50). To deal with the first problem, one can formally derive an equation of motion for a state vector in the Hilbert space that somewhat resembles a wave-function dynamics [82–84] and scales as *N* although its physical interpretation is slightly different [85,86]. However, the state vector follows a stochastic dynamic and therefore one needs to average over the realizations of the stochastic noise, balancing the computation gain of the reduced dimensionality. To deal with the second problem, we have two ways: we can neglect completely the history of the system, or retain part of it. In the Markov approximation [78–80] one arrives to the so-called Lindblad equation [87]

$$\frac{d\rho\_S}{dt} = -i\left[H\_{\rm S}\rho\_S\right] + \frac{\lambda^2}{2}(V\_{\rm S}^\dagger V\_{\rm S}\rho\_S + \rho\_S V\_{\rm S}^\dagger V\_{\rm S} - 2V\_{\rm S}^\dagger \rho\_S V\_{\rm S}).\tag{52}$$

Lindblad proved that this equation is the most general master equation of the Markov's type that preserves trace, positiveness and hermiticity of the density matrix *ρS* at each time steps up to second order in *λ*. A second approach follows from the observation that up to second order in *λ*, *ρS*(*t*) = *ρS*(0). Then, up to the same order of approximation, one replaces *ρ*(*<sup>t</sup>* ) with *ρ*(*t*), in Equation (50), i.e.,

$$M(t) \approx \int\_0^t dt' \mathbb{C}(t, t') e^{-iH\_S(t - t')} V\_S \rho\_S(t) e^{iH\_S(t - t')}.\tag{53}$$

This operator is now local in time, although one needs to calculate it at each time step. In the absence of any magnetic field, the bath dynamics satisfies time-reversal symmetry, and this is usually sufficient for *<sup>C</sup>*(*<sup>t</sup>*, *t* ) = *C*(*t* − *t* ), especially if *ρB*(0) is the equilibrium statistical density matrix of the

bath *ρ<sup>B</sup>*,*equ* = exp(−*βHB*) where *β* = 1/*kBT* and *T* the bath temperature, we can separate this integral and arrive at the Redfield master equation [80,88].

To describe thermal transport within OQS one needs to couple two environments, kept at different temperature, locally to the system

$$\frac{d\rho\_S}{dt} = -i\left[H\_{S\prime}\rho\_S\right] + \mathcal{L}\_L[\rho\_S] + \mathcal{L}\_R[\rho\_S].\tag{54}$$

Here, L*L* and L*R* describe the left and right environment, respectively. A possible nano-junction attached to two environments is shown in the upper panel of Figure 5. Furthermore, in Figure 5a one sees the voltage drop over the device induced by the applied temperature gradient, Δ *T*, due to the environmental coupling. One can observe a linear-response regime for small Δ *T*, a regime of rapid rise in Δ *V* and for large temperature gradients a region where the voltage drop has reached saturation due to the finite size of the system. The inset of Figure 5 shows the thermopower *S* = −d(<sup>Δ</sup> *V*)/d(Δ *T*) that presents a maximum in response to the thermal gradient at Δ *T* ≈ 0.25 [a.u.].

**Figure 5.** Upper Panel: Nanostructure connected locally to two reservoirs kept at different temperature. Below: Voltage drop, Δ*V* as a function of the temperature gradient, Δ*T*. The inset shows the thermopower *S* = −d(<sup>Δ</sup>*V*) d(Δ*T*) . Reprinted (adapted) with permission from [89]. Copyright (2009) American Chemical Society.

Note that all the shown quantities correspond to the steady-state solution of the master equations, where the long-time limit has been reached. In general, the OQS approach is not limited to this regime and can also be used to study time-dependent phenomena in nanoscale devices beyond linear response [78,90,91].

For the purposes of this review, it is relevant to point out that the quantum system is in general a device made of interacting electrons and ions. This, as we have seen (Section 3.1.1), is an incredibly complex problem and the coupling with the external environment does not make it simpler. Fortunately, we can use the theory of open quantum systems with (time-dependent) DFT to extract from a non-interacting open-quantum system information about the dynamics of the interacting one [58,83,92–95]. More surprisingly, it has been shown that the DFT for open-quantum system can be constructed in such a way that the KS system can be made closed. This means that the effects of the external environment can be included in the KS potentials [94,95]. However, it is not clear how one could effectively build the required KS potentials which would be depending on the coupling operators between the system and the environment. These difficulties have so far hindered a full development of the theory and its routine application for the investigation of transport in complex devices and physical conditions.

### *3.4. Influence of Decoherence onto Thermoelectrical Transport*

The observation or measurement process acting on a classical object does not influence its physical properties. However, when entering the nanoscale world and the quantum realm one needs to revisit some classical concepts. In the smallest nanodevices, electrons are usually set up in coherent superpositions, and a global measurement would destroy most of their coherence changing the dynamical response. Decoherence does not only originate from the measurement by a macroscopic observer, but it also occurs when the quantum system interacts locally with another quantum system. This effect of local quantum observation covers a variety of situations, such as, e.g., local electron–phonon coupling and continuous [96] or frequent [97] quantum non-demolition measurements. It has been shown that decoherence influences the efficiency of energy transport in biological [98–100] and molecular devices [100–102] and is responsible for the decoupling of the system from the environment via the Quantum Zeno effect [103,104].

While standard static transport approaches such as the BTE or LB fails in catching up with these dynamical effects, methods such as TDDFT or OQS approaches allow for the time-resolved study of electron transport and energy dissipation. Additionally, methods describing the dynamics within a density matrix formalism are suitable to study the role of coherence in nanoscale transport devices. Indeed, by modeling decoherence as an additional environment within a master equation approach, one can include it into a consistent thermodynamic formalism [105],

$$\mathcal{L}\_{\rm D}[\rho\_{\rm S}] = \lambda\_{\rm D} \left[ 2|\beta\rangle\langle\beta|\rho\_{\rm S}|\beta\rangle\langle\beta| - |\beta\rangle\langle\beta|\rho\_{\rm S} - \rho\_{\rm S}|\beta\rangle\langle\beta| \right]. \tag{55}$$

Here |*β* is the ket-vector representing the state of the spatial region where decoherence takes place. This bath can change the coherence of the system and has; in contrast to classical reservoirs that are in a thermal ensemble state, no temperature is associated with it [106]. This formalism has been applied to a ratchet-like device shown in Figure 6a [105].

By changing the on-site energy levels on the parallel horizontal branches of the device (graphically indicated by the size of the spheres), a spatial asymmetry is introduced. In the upper branch the on-site energies increase from left to right in equal proportion, while in the lower branch this is reversed. Therefore, the device acts as a quantum ratchet by the presence of two rectifiers on each branch in opposite direction. This directional transport device driven by thermal and quantum fluctuations has a preferred electronic current direction, in this example clockwise. In the ratchet, a hot ( *H*) and cold ( *C*) reservoir introduce a thermal gradient in the device while at the same time the decoherence bath ( *D*) is acting at site *β*

$$\frac{d\rho\_S}{dt} = -i\left[H\_{\rm S\nu}\rho\_S\right] + \mathcal{L}\_H[\rho\_S] + \mathcal{L}\_\mathbb{C}[\rho\_S] + \mathcal{L}\_D[\rho\_S].\tag{56}$$

While in this set-up energy is exchanged via the baths, particle current is then confined inside the device. Figure 6b shows the electronic current at steady state for the upper branch. Please note that by charge conservation, the current in the lower branch is exactly the same but flowing in the opposite direction. Here, red indicates a positive current, from left to right in the upper branch. If no decoherence is applied ( *λD* = 0), the current is flowing in the clockwise direction through the ratchet. Increasing the decoherence, the ring current decreases until its direction gets reversed. It has been found that the local quantum observer cannot only control the particle current but also energy currents in direction and strength inside the device [105]. This demonstrates that in thermoelectric nanodevices the current and heat flows are not only dictated by the temperature and potential gradients, but can also be manipulated by the external control of the coherence of the device. This effect is illustrated in Figure 6c. When looking at the whole picture the water flows uphill, however, when we only observe current flow in the circle (and ignore the rest of the illustration) it seems the water flows down the channel. This apparent paradox mimics the coherent superposition of two quantum states (water flowing up/down). The observation process in specific parts of our system can tune between these two quantum states and hence change the 'physical response of the nanodevice' in a controlled way.

**Figure 6.** Influence of quantum decoherence on the thermoelectric flows in a quantum ratchet shown in (**a**). (**b**) The electrical current in the upper branch can be seen. The current can change direction as a function of both *λD* and Δ*T*. (**c**) Artistic illustration (copyright K. Aranburu) of the role of decoherence in a nanodevice: When observing inside the black circle, it appears the water flows down the channel, instead, by looking at the whole painting the water actually flows uphill. This apparent paradox mimics the coherent superposition of two quantum states (water flowing up/down). By observing at specific parts of our system one is able to tune between these two states and hence change the 'physical response of the nanodevice' in a controlled way. Parts (**<sup>a</sup>**,**b**) are adapted from [105]. Copyright (2017) SpringerNature.
