Time-Dependent Dephasing and Quantum Transport

The investigation of the phenomenon of dephasing assisted quantum transport, which happens when the presence of dephasing benefits the efficiency of this process, has been mainly focused on Markovian scenarios associated with constant and positive dephasing rates in their respective Lindblad master equations. What happens if we consider a more general framework, where time-dependent dephasing rates are allowed, thereby, permitting the possibility of non-Markovian scenarios? Does dephasing-assisted transport still manifest for non-Markovian dephasing? Here, we address these open questions in a setup of coupled two-level systems. Our results show that the manifestation of non-Markovian dephasing-assisted transport depends on the way in which the incoherent energy sources are locally coupled to the chain. This is illustrated with two different configurations, namely non-symmetric and symmetric. Specifically, we verify that non-Markovian dephasing-assisted transport manifested only in the non-symmetric configuration. This allows us to draw a parallel with the conditions in which time-independent Markovian dephasing-assisted transport manifests. Finally, we find similar results by considering a controllable and experimentally implementable system, which highlights the significance of our findings for quantum technologies.


I. INTRODUCTION
Dephasing assisted transport means energy currents enhanced by dephasing [1,2]. This implies that the open system dynamics may surpass the correspondent unitary evolution in terms of transport efficiency. On the one hand, this defied the notion that, in general, the presence of noise tends to jeopardize the efficiency of tasks performed by quantum systems [3]. On the other hand, it helped us to understand energy transport behavior in quantum systems subject to heavily noisy conditions in harsh natural environments, which nonetheless shows an outstanding ability to effectively transfer energy. A paradigmatic example is the widely studied Fenna-Mathew-Olson (FMO) complex, a structure present in green sulphur bacteria which channels the energy captured from solar light to a reaction centre [4][5][6][7][8][9]. As well as this, the comprehension of dephasing assisted transport is of central importance for quantum technologies. Indeed, the possibility of exploiting it to achieve improved efficiencies is very appealing from the point of view of practical implementations, principally for quantum technology applications including controlled quantum systems [10][11][12][13][14].
The theoretical studies of dephasing assisted transport have been mainly focused on time-independent interaction between the system and environment [1,15,16]. Therefore, investigations of time-dependent dephasing in a transport scenario that includes more general Markovian as well as non-Markovian evolutions have the potential to drive new applications in the context of quantum technologies [17]. Furthermore, in the last years there has been a great interest in the fundamental and practical aspects of non-Markovianity [12,[18][19][20][21][22][23][24][25]. With the tools resulting from these studies and the experimental advances that have been reported, it is natural to envisage new possibilities to exploit such systems in the context of quantum transport. In this work, we will study how the presence of time-dependent dephasing in a chain of coupled two-level systems affects quantum transport efficiency in Markovian and non-Markovian scenarios. In doing so, we tackle a relevant question in the field of open quantum systems, which is the impact of timedependent scenarios on quantum transport. We will be using the fact that important examples of non-Markovian evolutions can be characterized by Lindblad-like master equations for which the time dependent decoherence rate achieves negative values [17,[26][27][28][29].
This paper is organized as follows. First, we review the canonical form of the Lindblad-like master equations and the characterization of non-Markovianity via master equations in section II. Then, we describe the transport model in section III. In section IV, we present our results and analyse transport efficiency in some time-dependent dephasing scenarios, and extend the analysis for results obtained in the context of a controlled quantum system in section V. In section VI, we present our conclusions.

II. CHARACTERIZING TIME-DEPENDENT NON-MARKOVIAN EVOLUTIONS
Time-local master equations [30,31] can be expressed in a Lindblad-like form aṡ with a unique set of functions γ k (t), not necessarily positive for all times [29]. Here, d is dimension of the state space, H(t) is a Hermitian operator, andL k (t) constitutes an orthonormal basis of traceless operators, i.e., Since any time-local master equation can be written in this canonical form, in which each γ k (t) is uniquely de-arXiv:2102.10466v1 [quant-ph] 20 Feb 2021 termined, it turns out that Eq.(1) may be used to characterize non-Markovianity [29]. In fact, γ k (t) ≥ 0 means Markovianity, since it is equivalent to the divisibility of the map into completely positive evolutions [26][27][28]32]. Therefore, a strictly negative value of γ k (t), for some k and at any instant of time t, indicates non-Markovianity. The fact that each γ k (t) is unique in Eq.(1) motivated the use of as a indicator of non-Markoviany in the channel k, and its integration in time as a quantifier of the total amount of non-Markovianity of a given channel k in an interval of time from t to t [29]. In general, γ k (t) must satisfy certain constraints for a completely positive evolution. For instance, consider a master equation for a two-level system given bẏ where σ i are Pauli matrices (σ 1 = σ x , σ 2 = σ y , σ 3 = σ z ), and H(t) is Hermitian. Complete positivity of the map, in the interval from 0 to t, is garanteed if the following set of conditions are fulfilled [33] Γ j + Γ k ≤ 1 + Γ l , for all permutations j, k, l of 1, 2, 3 where Γ j ≡ exp(− t 0 ds[γ k (s) + γ l (s)]). Let us illustrate it with a simple case where γ 1 (t) = γ 2 (t) = 0 and γ 3 (t) = γ(t), i.e., It is straightforward to show that t 0 γ(s)ds ≥ 0 is the requirement for the map to be completely positive. This master equation will be important for our investigation of the phenomenon of non-Markovian dephasing assisted transport in the rest of the paper. An example is given by γ(t) = sin(νt), where ν is integer and γ ≥ 0 [34]. Such functions satisfy the aforementioned condition and therefore the map is CP for all t.

III. THE MODEL
We consider a linear chain of N two-level systems in a first-neighbor coupling model, whose Hamiltonian is given by ( = 1) where σ + i is the operator causing transition from ground to excited state in site i, σ − i = (σ + i ) † , σ z i and ω i are the Pauli z operator and the energy associated with ith site, respectively, and λ i is the coupling constant between sites i and i + 1. This model has been extensively used to describe quantum transport, and this kind of interaction can be implemented, for instance, in the context of trapped ions and circuit QED [25,35].
In turn, the chain is considered to be locally coupled to incoherent energy sources, responsible for incoherent injection and extraction of energy. More specifically, we consider energy injection at site 1 and extraction at site k, where 2 ≤ k ≤ N . This situation is described by the following terms, to be added to the master equation where κ inj (κ ext ) describes the rate of injection (extraction) of energy into (out) the chain. In order to simplify the notation, we omitted the time-dependence of ρ(t) in Eq. (9). From now on, we will adopt this simplified notation.
Finally, we consider that each site is also subjected to local dephasing. This assumption of local coupling to the environment is reasonable for weak intercoupling strength between the sites of the chain when compared to the local frequencies [36][37][38][39][40][41]. For the sake of simplicity, we will assume that each site is subjected to equivalent dephasing environments. Therefore, the total dephasing to which the chain is subjected is given by Then, non-Markovianity is the result of γ(t) assuming negative values. Finally, the total master equation representing the evolution of the system will reaḋ For the investigation of transport efficiency, we will consider the stationary value of the rate of variation of the total number operatorN = i σ + i σ − i . In a broad sense, it can be called exciton current. In the stationary state, one finds Tr As a figure of merit for transport efficiency, we then consider is the asymptotic population of the extraction site. To be more specific, we will be evaluating the rescaled current JN ≡ (κ ext N ) −1 JN .

IV. TIME-DEPENDENT DEPHASING ASSISTED TRANSPORT
We start our analysis by considering time-dependent dephasing models with sinoidal time dependence. We focus on two specific situations, called symmetric and non-symmetric configurations, having in mind the case N = 7 as a benchmark for the Markovian case [15]. In the non-symmetric configuration, the 5th site is the extraction site, what breaks inversion symmetry. In the symmetric configuration, the extraction site is the 7th site. Chains of different sizes can also be studied and they present similar behaviors to the ones presented here. In both cases, the chain has uniform frequencies ω i = ω and inter-site couplings λ i = λ, and the injection site is always the first site. In the symmetric configuration, the extraction site is on the other tip of the chain, i.e. the last site. All other choices for extraction site will lead to non-symmetric configurations. As shown in Ref. [15], Markovian dephasing-assisted transport manifests only in the non-symmetric configuration. We investigate what happens when non-Markovian dephasing shows up in these configurations. To construct a scenario, we set ω i = ω, λ i = λ = 0.1ω, and κ ext = κ inj = 0.01ω for all simulations, i.e. all frequencies and couplings are set in units of ω.

A. Non-symmetric configuration
In Fig. 1, we plot the currentJN as a function of γ ≥ 0 for the time-dependent dephasing model γ(t) = γ sin(νt), and different values of ν in the non-symmetric configuration. We also consider the average of these three sine functions, for which ν = 0.3, 1, 4. This model happens to be non-Markovian for any finite value of the positive constant γ. The Markovian case corresponding to γ(t) = γ is also plotted as a benchmark. The first thing to be noticed is that dephasing-assisted transport manifests in both cases: Markovian and non-Markovian. This corresponds to the first portion of the curves where the current increases with γ. Regardless of being Markovian or not, there is always an optimal value of γ above which dephasing becomes detrimental. Notwithstanding, we see that the non-Markovian cases are more efficient than their Markovian counterpart for higher dephasing magnitudes γ.
In Fig. 2, we consider the currentJN for another model, for which γ(t) = γ+γ 0 sin(t) in the non-symmetric configuration, with γ 0 = 1. The resulting dynamics is non-Markovian for 0 < γ < 1. One can also see the timeindependent Markovian benchmark in the same plot. As a glimpse of how rich the transport scenario is in the presence of time-dependent dephasing, this model does not present efficiency enhancement by dephasing. Compared to the Markovian case for γ = 0, i.e. closed system dynamics, the case with γ(t) = γ + sin(t) is always less efficient. This is in clear contrast to the model considered before. However, the present model shows an interesting non-monotonic behavior with γ, and it also turns out to be more efficient than the time-independent Markovian counterpart for higher values of γ. FIG. 2: Non-symmetric configuration -currentJN as a function of γ/ω for γ(t) = γ + γ0 sin(t). The dotted red vertical line corresponds to γ = 1. The system is decreasingly non-Markovian in the interval 0 < γ < 1. For γ ≥ 1, the system is Markovian since we have γ(t) ≥ 0 for all t. The gray curve corresponds to γ(t) = γ.

B. Symmetric configuration
Now, we focus on the symmetric configuration, and once again plot the currentJN as a function of γ for γ(t) = γ sin(νt), where ν = 0.3, 2, 4. As before, we also plot the case with the average of the aforementioned sine functions, what guarantees a fair comparison with the other cases. The Markovian case corresponding to γ(t) = γ is also plotted and shows a monotonic behavior as γ is increased. In other words, there is no Markovian dephasing-assisted transport in the symmetric case, in agreement with Ref. [15]. For each non-Markovian curve shown in Fig. 3, we also have that non-Markovian dephasing assisted transport does not manifest, as the maximum current is reached for γ = 0. Nevertheless, it is remarkable to see that the non-Markovian cases becomes once again more efficient than their Markovian counterpart as γ is increased.
In Fig. 4, we consider once again the model given by γ(t) = γ + γ 0 sin(νt), with γ 0 = 1 in the symmetric configuration. As we can see, a non-monotonic behaviour is also observed in this case, and, by comparing it to the the benchmark, we see that it can also help efficiency regardless of being Markovian (γ < 1) or not (γ ≥ 1).

C. Spread of occupations and efficiency
Next, we seek to analyse how the spread of occupations correlates with the current maximum in the timedependent dephasing scenarios presented above. We consider the spread of occupations ∆ n [15], with n i = p i (∞), where n k is the population of the extraction site k, n k = p k (∞). The maximum of this quantity is associated with a minimum spread of the occupations. A correlation between the maximal of ∆ n and the maximum of the current is verified in Ref. [15] for several time-independent Markovian cases. Here, we certify that, for the timedependent non-Markovian cases studied above, the same tendency is verified: ∆ n is maximum when the current is maximum, as shown in Figs. 5 and 6, which shows plots of ∆ n and the current as a function of γ. These results suggest that this quantity is an indicator of optimal transport scenarios in the more general time-dependent and non-Markovian dephasing picture.

V. EXAMPLE: CONTROLLED QUANTUM SYSTEM
In the scope of controlled quantum systems, timedependent dephasing, including non-Markovian evolutions, can be introduced and externally controlled [12,[18][19][20][21]. Usually one can achieve it through controlled auxiliary systems. Specifically, a model in the context of nuclear magnetic resonance (NMR) experiments, where a Ising-like interaction takes place between two spin 1/2 systems, is studied in Ref. [18]. One of these two-level systems is considered to be the system of interest, and the other is is seen as part of the environment, providing a structured bath. The strength of the coupling between the system and the environment is given by a parameter J, and θ is a parameter which gives the state in which the environment is initialized, before the interaction. It turns out that the parameters J and θ are controllable in the NMR experimental realization. In particular, the following superoperator can be engineered for any site i of the chain, such that the master equation describing the system's state ρ is given by [18] where γ i (t) = γ i + πJ sin 2 (2θ) sin(2πJt) 3 + 2 cos(4θ) sin 2 (πJt) + cos(2πJt) , is a time-dependent dephasing rate, and s i (t) = 2πJ cos(2θ) 3 + 2 cos(4θ) sin 2 (πJt) + cos(2πJt) , is an environment-induced time-dependent energy shift. As mentioned before, in Eqs. (14) and (15), J and θ are fully controlled parameters. First of all, it is worth studying the behaviour of the function γ i (t) in Eq. (14), which is an odd and periodic function satisfying the condition for a completely positive evolution for any value of γ i ≥ 0 as discussed in section II. In all the plots in this section, we will consider J = 1. In Fig. 7, we plot γ i (t) for several values of the parameter θ while keeping γ i = 0. We can see that, for γ i = 0 the system is always non-Markovian, and its non-Markovianity increases as θ increases in the interval [0, π/2]. For simplicity, we assume that all γ i = γ so that  all γ i (t) are the same. Thus, the whole chain will be subjected to the following total master equation, which also takes into account the coupling to the incoherent energy sources responsible for injection and extraction of energy, as described before,

A. Non-symmetric configuration
We now focus on the model with time-dependent dephasing as described by Eqs. (14), (15) and (16). First, we consider γ = 0 in Eq. (14). In Fig. 8, we have the cur-rentJN plotted as a function of θ, see the solid green line. We see that non-Markovian dephasing assisted transport happens, as the maximum of the current is associated with a value θ = 0. This is similar to the behavior discussed before and illustrated in Fig. 1, but here θ is the parameter controlling the non-Markovian dephasing. In other words, by increasing θ, one increases the presence of non-Markovian dephasing in the system's evolution.
The effect of increasing the positive contribution γ is shown in Fig. 9 for fixed J and θ (J = 1 and θ = 0.52, for which γ(t) is plotted in Fig. 7), where the solid green line is the plot of the current as a function γ in the non-symmetric configuration. The system is initially non-Markovian and becomes Markovian for the value of γ ≈ 1.17 correspondent to the dotted red line. For γ 1.17, the system is Markovian. We see that the increase in the Markovian contribution, γ, cannot lead to an increase in the current,JN . We note that the same effect -the decrease ofJN as γ increases -is observed for other values of J and θ. Therefore, the decrease in the non-Markovianity of the system by increasing γ jeopardizes transport efficiency. The system is decreasingly non-Markovian in the interval 0 < γ 1.17. For γ 1.17, the system is Markovian since we have γ(t) > 0. The green line represents the non-symmetric case, while the dashed coral line represents the symmetric one.

B. Symmetric configuration
We consider the case γ = 0 in Eq. (14) in the symmetric configuration. In Fig. 8, we have the current JN plotted as a function of θ, see the dashed coral line. We see that non-Markovian dephasing assisted transport does not happen, as we have a monotonic behavior of the current with θ. This behavior is similar to the cases discussed before for Fig. 3, where we also have a monotonic behavior for the current in all the cases.
Next, we study what happens when the positive contribution γ is increased, by taking J = 1 and θ = 0.52 once again, but now in the symmetric configuration. In Fig.  9, the dashed coral line shows the current as function of γ. As in the non-symmetric case, we see a monotonic decreasing behavior of the currentJN as γ decreases. As before, the non-Markovian system becomes Markovian for γ ≈ 1.17, indicated by the dotted red line in Fig. 9. As in the non-symmetric case, therefore, non-Markovian scenarios are shown to be associated with a greater transport efficiency.

VI. CONCLUSION
The investigation of the influence of time-dependent dephasing rates γ(t) on the efficiency of quantum transport is a very relevant open problem in the context of open quantum systems research, notably in non-Markovian scenarios for which γ(t) reaches strictly negative values. Here, we provided a systematic investigation of this phenomenon for a chain of coupled two-level systems, which is, in turn, locally coupled to incoherent sources of energy, a scenario which accounts for energy injection and extraction in and out of the chain. An exciton current is then established and evaluated when the system is in the stationary state. Specifically, the models treated here are characterized by complete positive maps associated with time-dependent dephasing rates described by linear combinations of sine functions and constants in the canonical representation of the corresponding master equation. Based on that, we studied the behavior of the exciton current in the non-symmetric and symmetric scenarios. We find that the phenomenon of non-Markovian dephasing-assisted transport occurs in the non-symmetric cases, thereby establishing a parallel with the time independent Markovian cases investigated elsewhere [15].
As a final remark, one can also investigate the so-called "maximally non-Markovian evolution" [42], to find that different degrees of non-Markovianity in that model do not change the efficiency of quantum transport. This indicates that the generalization of non-Markovian dephasing-assisted transport, beyond the models studied here, is not straightforward, and certainly deserves to be further investigated. We hope our work will serve as a motivation for further advances related to this interesting problem.