A Knob for Markovianity

We study the Markovianity of a composite system and its subsystems. We show how the dissipative nature of a subsystem's dynamics can be modified without having to change properties of the composite system environment. By preparing different system initial states or dynamically manipulating the subsystem coupling, we find that it is possible to induce a transition from Markov to non-Markov behavior, and vice versa.

The theory of open quantum systems plays a central role in the description of realistic quantum systems due to unavoidable interaction with the environment. As it is well known, the system-environment interaction can lead to effects of energy dissipation and decoherence [1], establishing a major challenge in the development of modern technologies based on quantum coherence [2]. Due to its fundamental character and practical implications, the investigation of dissipative processes has been a subject of vigorous research, where the standard approach assumes a system-environment weak coupling and a memoryless quantum dynamics (the Born-Markov approximation). Under such assumptions, the system dynamics is determined by a quantum Markovian master equation, i.e., a completely positive quantum dynamical map with a generator in the Lindblad form [1,3].
Although the Markovian approach has been widely used, there is a growing interest in understanding and controlling the non-Markovianity. In quantum metrology, for example, entangled states can be used to overcome the shot noise limit [4] in precision spectroscopy, even in the presence of decoherence [5]. However, as suggested by the finds presented in [6], higher precision could be achieved in a non-Markovian environment, since a small Markovian noise would be enough to restore the shot noise limit. The non-Markovian dynamics also plays important role in quantum biology [7], where the interaction with a non-Markovian environment could be use to optimize the energy transport in photosynthetic complexes [8], and in superconducting qubits [9] and quantum dots [10], whose main source of decoherence can be of that nature. Furthermore, as pointed out recently in some studies involving quantum key distribution [11], quantum correlation generation [12], optimal control [13], and quantum communication [14], the use of non-Markovian dynamics could bring an advantage compared to Markovian ones.
This scenario has motivated studies in order to characterize and quantify the non-Markovian aspects of the time evolution of an open quantum system. However, unlike the classical case, the definition of non-Markovianity in the scope of quantum dynamics is still a controversial issue. For example, Breuer, Laine and Piilo (BLP) [15] have proposed a measure for non-Markovianity using the fact that all completely positive trace preserving (CPTP) maps increase the indistinguishability between quantum states. In a physical perspective, a quantum dynamics would be non-Markovian if there would be a temporary back-flow of information from the environment to the system. On the other hand, for Rivas, Huelga and Plenio (RHP) [16], a quantum dynamics would be non-Markovian if it could not be described by a divisible CPTP map. Formally, for such cases, one could not find a CPTP map Φ : ρ(0) → ρ(t) = Φ(t, 0)ρ(0), describing the evolution of the density operator ρ from time 0 to t, such that Φ(t + τ, 0) = Φ(t + τ, t)Φ(t, 0), where Φ(t + τ, t) and Φ(t, 0) are two CPTP maps. Therefore, the indivisibility of a map would be the signature of a non-Markovian dynamics. Those two different concepts of non-Markovianity are not equivalent [17]: although all divisible maps are Markovian with respect to BLP criterion, the converse is not always valid [18].
In this Letter, we put forward the idea of how one could manipulate the Markovianity nature of a dissipative subsystem, by exploiting features of being a part of a composite system. For that, we study the dynamics of interacting two-state systems (TSS) coupled to a common reservoir. By means of changing the composite system initial state and/or the TSS couplings, we show that it is possible to modify in situ the characteristics of the subsystem dissipation, enabling one to induce a transition from Markovian to non-Markovian dynamics, and vice versa. Moreover, we observe the possibility of having different behaviors for the composite and subsystem, even when they are coupled to a common environment. In this sense, we show that the dynamics of a subsystem can be Markovian even when the composite system displays a non-Markovian behavior.
We choose an exactly soluble analytical model that is capable of presenting the physics we want to exploit from dissipative composite systems. Therefore, our starting point is the dephasing model for two interacting twostate systems (2-TSS) with diagonal Pauli matrix and S z ≡ σ z 1 + σ z 2 . The bath of oscillators, introduced by the canonical bosonic creation and annihilation operators b † k and b k , is characterized by its spectral density J(ω) ≡ k |g k | 2 δ(ω − ω k ) [19], and responsible for raising a nonunitary evolution for the 2-TSS. Since [σ z i , H] = 0, the populations of the eigenstates of (σ z 1 , σ z 2 ) are constants of motion and the coupling with the environment solely induces random dephasing between any superposition of those eigenstates. The (2-TSS)-bath time evolution operator can be readily determined as denotes the density matrix of the 2-TSS plus bath, then ρ SB (t) = U (t, 0)ρ SB (0)U † (t, 0). Regarding the (2-TSS)bath initial correlations, the initial state ρ SB (0) is here onwards assumed to be separable, i.e., ρ where [ , ]({ , }) denotes the standard (anti)commutator, and γ(t) ≡ dω J(ω) ω sin ωt coth ω 2k B T is the timedependent dephasing rate. Since no approximation was done whatsoever, it is worth noting that Eq. (3) constitutes a genuine CPTP quantum dynamical map.
The master equation Eq. (3) has a very suitable form for the analysis of quantum Markovianity, since it can be directly compared with the well-known Lindblad theory for open systems [1,3]. Indeed, if γ(t) ≥ 0 for all t ≥ 0, the time-local master equation describes a divisible CPTP map. Therefore, under this condition, the dynamics would fall in the class of problems considered as paradigm of quantum Markov processes [15,16,18].
The simplest example one can find for the system Hamiltonian Eq. (1) that has γ(t) ≥ 0, ∀t ≥ 0, is the case of an ohmic bath, i.e., J(ω) = ηωe −ω/ωc [31]. Indeed, as depicted in Fig. 1(a), for any bath temperature T , the dephasing rate satisfies the condition of being nonnegative for each fixed t ≥ 0. Consequently, the 2-TSS dissipative dynamics would be categorized as Markovian.
Another important case happens when the environment seen by the system of interest presents a pronounced peak (often referred as Lorentzian peak [20]) at a characteristic frequency Ω. Relevant examples are superconducting qubits coupled to readout dc-SQUIDS [9], electron transfer in biological and chemical systems FIG. 1. The dephasing rate γ(t) (a-b) and trace distance Eq. (7) (c-d) as a function of time for the system Hamiltonian Eq. (1) in an Ohmic (a, c) and Lorentzian (b, d) environment. Depending upon the bath temperature T , it is found non-negative dephasing rates for all t for both spectral densities, indicating regimes of Markov behavior for the 2-TSS dynamics. The trace distance is performed for the TSS reduced density matrix for two TSS-TSS coupling strengths J, considering a bath temperature T where the 2-TSS dephasing rate has been found to be always positive, and orthogonal initial states given by The non-monotonic behavior is a signature of non-Markov process for the single TSS dynamics. Physical parameters used for this plot: Ω = l = 0.01ωc [21] and semicondutor quantum dots [22], to name just a few. Indeed, in general one should expect such physics in cases where the energy scale of the quantum detector used to determine the state of a system of interest is comparable to its own energy scale. Because of the quantum detector also suffers from dissipative processes, the resonance system-detector constitutes an efficient channel for decoherence [23]. Fig. 1(b) shows the dephasing rate assuming a Lorentzian shape for the bath spectral density [20,21]. As one can observe, for the case which the resonance width l is of the same order of the frequency peak Ω, the dephasing rate can present negative values (solid curve) or be a positive-valued function (dashed and dot-dashed curves) depending upon the bath temperature. Thus, the composite dissipative dynamics would be Markovian as long as the thermal energy scale is comparable or larger to the resonance parameters (dashed and dot-dashed curves), i.e., k B T Ω ∼ l.
Thus far, we have only been concerned with characterizing the dissipative dynamics of the composite system, i.e., the open 2-TSS. What we now want to address is whether or not the single TSS dissipative dynamics could differ in behavior from the one observed for the composite system. In fact, one could envision that, due to its interaction to the other TSS (here onwards labeled as the auxiliary TSS), a single TSS would be coupled to a structured bath (auxiliary TSS+environment), which would play the role of an effective bath, and could induce a different nature for the dissipative process. If so, it might be the case that it would be possible changing in situ the nature of such structured bath by means of varying the TSS-TSS coupling J and/or the initial state of the auxiliary TSS.
As matter of fact, the approach of considering the subsystem's environment as comprised of a common environment plus the rest of the composite system has been employed long ago. Indeed, regarding to the characterization of an effective dissipative mechanism, it has been successfully applied to the class of problems mapped onto a TSS coupled to a harmonic oscillator in the presence of a Markov bath, where perturbative [20,21], nonperturbative [24], spectral density series representation [25] and semi-infinite chain representation [26] techniques have been proposed to describe such an effective dynamics in several contexts. Nevertheless, neither systematic methods have been taken to investigate their Markovianity, nor a TSS-TSS composite system was analyzed, which tunability seems more natural for manipulating the dissipative dynamics nature.
It follows from Eq. (4) that the master equation for the single TSS reduced density matrix readṡ It is worth noticing the Lindblad structure of the master equation Eq. (5), whereH 1 (t) andγ(t) play the role of effective single TSS Hamiltonian and dephasing rate, respectively, which are manifestly dependent on the composite system initial state condition. Moreover, since β depends only on the initial state ρ s (0) and the TSS-TSS coupling J, both the unitary part of Eq. (5) as well as the extra term in its dephasing rate are not influenced by the systembath coupling. Consequently, the bath and auxiliary TSS contributions for each term of Eq. (5) can be identified and quantified.
Two results immediately become apparent: i) as expected, if the TSS-TSS coupling is zero (J = 0 → dβ(t)/dt = 0 →J = 0 andγ = γ), the single TSS dissipative dynamics is completely enforced by the environment; and ii) if the auxiliary TSS initial state is set in an eigenstate of σ z , one finds that Those results just highlight the fact the auxiliary TSS dynamics will be locked in that initial state, since σ z is a constant of motion, and therefore the interacting term Jσ z 1 σ z 2 will only play the role of a fixed external field for the single TSS, which dissipative dynamics would follow the process imposed by the environment. Thus, for such cases, both the composite and single TSS dissipative dynamics would have the same behavior.
In spite of the two cases discussed above, in generalγ is neither determined only by the environment dephasing rate γ nor is a positive-valued function. Therefore, with regard to the concordance between BLP and RHP measures, it is not possible to infer from the master equations Eqs. (3) and (5) if the single TSS dissipative dynamics will follow or not the same behavior as the one observed for the composite system. A turn around to this problem comes from the perspective of looking at the distinguishability of quantum states [1]. An important tool that has been used as a measure for the distinguishability of quantum states is the trace distance between two quantum states ρ and σ where |A| ≡ √ AA † identifies the positive square root of AA † [2]. Among several other properties [2], the trace distance has the feature that under CPTP maps Φ(t, 0) its value cannot increase beyond its initial value, i.e., D(ρ a (t), ρ b (t)) ≤ D(ρ a (0), ρ b (0)), where ρ a,b (t) = Φ(t, 0)ρ a,b (0). Because of that, it has been put forward by Breuer, Laine and Piilo (BLP) that the trace distance could also be used to set a definition of non-Markovian quantum dynamics [15,18]. That definition is based upon the idea that a Markovian dynamics has to be a quantum process where any two quantum states should become more and more indistinguishable as the dynamics flows, leading necessarily to a monotonic decrease of its trace distance. Thus, a non-monotonic behavior is interpreted as a back-flow of information from the environment to the system [32]. In an experimental perspective, the trace distance could be calculated by the tomography of the density operator, as it was recently done by Bi-Heng Liu et al. [27] using an all-optical experimental setup. For states Eq. (4), the trace distance can be easily determined as which time behavior is manifestly dependent on the auxiliary TSS initial state. For instance, panels (c) (Ohmic) and (d) (Lorentzian bath) of Fig. 1 show a representative initial condition, in which the auxiliary TSS is set in an equal superposition of σ z eigenstates, i.e., (|+ +|− )/ √ 2. The bath temperature is chosen such that both Ohmic and Lorentzian baths raise a Markovian process for the composite system. However, even though it asymptotically vanishes, for both cases the trace distance for single TSS states is non-monotonic, indicating a non-Markovian process. Moreover, this non-monotonic behavior implies that the TSS dynamics is indivisible [18]. Thus, the TSS dynamics is non-Markovian not only in terms of the backflow of information, but also with respect to the RHP criterion.
On the other hand, one can find a situation where the single TSS dynamics is Markovian even when the composite system presents a non-Markovian behavior. This unexpected scenario is shown in Fig. 2. The negative values assumed by the dephasing rate γ(t), panel (a), ensure that the 2-TSS dynamics is indivisible. Furthermore, as shown in panel (b), the trace distance can be found having a non-monotonic behavior. Therefore, the composite system conforms a non-Markovian quantum process with respect to the BLP and RHP criteria. However, since the effective rateγ(t) is always positive for this fixed initial condition ρ 2 (0), inset of panel (a), we have a divisible dynamics, implying that the trace distance is a monotonic function of time [18], as illustrated in panel (b). Consequently, the single TSS dynamics is a Markovian process in the standard way.
The examples brought here demonstrate that the presence of the auxiliary TSS not only has quantitative effects, but may also have, with regard to BLP and RHP criteria, a decisive impact on the nature of the TSS dynamics. Such influence can be made more explicit if one focus on cases where the composite initial state is a product state, i.e., ρ s (0) = ρ 1 (0)⊗ρ 2 (0). In fact, for those the extra term inγ, namely, γ aux (t) ≡ − β |β| 2 dβ * dt , can be written in the simple form which is a π/J periodic function of alternating sign, with amplitude γ max |. Furthermore, if one considers problems having the same ρ 2 (0), the figure of merit for both criteria becomesγ, since the monotonicity of the trace distance Eq. (7) is determined by dD/dt ∝ −γ. Thus, asγ ≡ γ +γ aux , it is clear that can be established a competition when setting the nature of the reduced TSS dynamics, which can be tailored through the knobs J and σ z 2 due to the presence of the auxiliary TSS. Indeed, except for the trivial case | σ z 2 | = 1, Eq. (8) shows that the term due to the coupling with the auxiliary TSS will induce an ad infinitum pattern of momentary loss and recurrence of the quantum coherence observed for the reduced TSS dynamics, which is a characteristic of the entanglement created between a tiny number of degrees of freedom. Thus, the coupling with the auxiliary TSS constitutes a channel, here reversible, for exchanging information. The reversibility of such channel is only possible here because of the diagonal nature of the interactions, which maintains constant the amplitude of γ aux . Consequently, the irreversibility of the dephasing process observed for the reduced TSS is only led by its direct coupling to the environment. Following that reasoning, one could expect that the same results would be found for the reduced TSS dynamics if an independent bath model were assumed, instead of the common environment model Eq. (1). As a matter of fact, as long as the diagonal nature of the interactions is preserved, the same results Eqs. (4)-(8) are obtained, with γ(t) reading the dephasing rate due to the individual bath coupled to the single TSS.
Observe that, if ρ 2 (0) is assumed a thermal state, our problem resembles very much with the one having a unique (structured) bath initially decoupled from the single TSS. Despite the several examples known for the TSS-harmonic oscillator problem [20][21][22][23][24][25][26], because of the different character between the spin and bosonic degrees of freedom, it seems not possible to assign an effective spectral density for the TSS-TSS case. Nevertheless, one can single out a spectral density due to the spin character of the structured bath. Following from Eq. (8), one finds that γ aux (t) = dω Jaux(ω) ω sin ωt = dω Jaux(ω) ω sin ωt exp − 2 J ω 2k B T , which leads to the spin component of the spectral density J aux (ω) ≡ (2J) 2 ∞ m=1 (−1) m+1 mδ(ω − 2Jm). Those finds reveal that the auxiliary TSS behaves as a mode filter for the natural 2J frequency and its harmonics, which are weighted by a Boltzmann factor of effective temperature T spin ≡ (2J/ 2 )T .
In summary, we have shown the possibility of manipulating the Markovianity of a subsystem of interest without having to change the common environment properties. By choosing an exactly soluble model, we illustrated how one can induce a transition from Markovian to non-Markovian dynamics, and vice versa, by means of changing the characteristics of a composite system. It is worth mentioning that the idea of manipulating the dissipative process of a subsystem is not model-limited, and that the particular model and protocol chosen could be implemented in superconducting qubits [28], trapped ions [29], and ultracold atoms in an optical lattice [30], for example.