Abstract
The question, whether an open system dynamics is Markovian or nonMarkovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a twolevel system interacting with an environment, which is also a heat bath, and consists of a large number of twolevel quantum systems. Our model can be seen as a close approximation to the ‘spin bath’ model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of timeindependent or timepolynomial coupling gives rise to nonMarkovianity. Also, we witness nonMarkovianity for certain parameter values of timeexponential coupling. Moreover, we study the transition from nonMarkovian to Markovian dynamics as we change the value of coupling strength.
Introduction
We rarely come across systems that are completely isolated from the surrounding world. Had it been the case, dealing with quantum mechanical systems would have been lot more easier. So, although arduous to deal with, real quantum systems are mostly open quantum systems – a system interacting with an environment. In these situations, information exchange between system and environment becomes an essential feature. Information that has been previously transferred to the environment may come back and affect the system, and this may appear as a memoryeffect on the system. When this information backflow from the environment is negligible we have a situation analogous to the discrete Markov process, where the instantaneous state of the system depends solely on the immediately previous step, the system dynamics is called memoryless or Markovian^{1,2}. On the other hand, when this information backflow affects the system significantly i.e. when some long past history of the system influences its present state, the system dynamics becomes retentive, and is called nonMarkovian.
In recent years, nonMarkovianity has been used as a resource in a number of information theoretic protocols, namely, channel discrimination^{3}, preserving coherence and correlation^{4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19} and retrieving quantum correlations in both quantum and classical environments^{16,20,21,22,23}. NonMarkovian effects also play important roles in areas ranging from fundamental physics of strong fields^{24,25} to energy transfer process of photosynthetic complexes^{26}.
Owing to its diverse applications, various aspects of nonMarkovianity are now being studied. Lately, researchers have been focusing on transition from nonMarkovian to Markovian dynamics^{27,28,29,30,31,32}. Some of them have dealt with bosonic bath of infinite or finite degrees of freedom, while some have considered a qudit system as the environment. But in all of these studies, systemenvironment interaction has been considered to be homogeneous in space, and the issue of nonMarkovian to Markovian transition in terms of systemenvironment coupling strength has not been addressed. Note that, nonMarkovian to Markovian transition is, in general, not a trivial issue, as in most cases finite dimensional environments give rise to nonMarkovianity.
In our study, we attempt to analyze the problem of whether a transition from nonMarkovianity to Markovianity can be engineered for the spin bath model^{33}. We particularly choose the spin bath model since it has wide ranging applications in simulating real physical scenarios^{33,34,35}. In our attempt, we face a serious difficulty in diagonalizing the spin bath Hamiltonian, either analytically or numerically, for larger number of spins in the environment. Although, analytic solutions do exist for constant coupling^{36} and some special forms of time dependent coupling^{37}, general solution for arbitrary forms of systemenvironment coupling of the spin bath Hamiltonian are hard to find. We therefore, try to circumvent the problem by choosing a simple model, which we argue, is a close approximation to the spin bath model for low temperatures. We choose an exchange type of interaction between a system qubit and individual environment qubits, where for each environment qubit the coupling can be chosen to be of different time dependent forms. But unlike the spin bath case, in our model, when the exchange interaction takes place between the system and a particular environment qubit, the rest of the environment qubits remain in a ground state; which also closely resembles the state of environment for low temperatures. As we will see in the paper, this approximation helps us to calculate and analyze nonMarkovian to Markovian transition for different types of systemenvironment coupling.
We present four scenarios here, for different forms of systemenvironment coupling: (i) the coupling is timeindependent and homogeneous over environment qubits, (ii) the coupling is timeindependent but inhomogeneous over environment qubits, (iii) the coupling is homogeneous over the environment but is timedependent, and (iv) the coupling is both timedependent and inhomogeneous. We find that cases (i) and (ii) always give rise to nonMarkovian system dynamics. For cases (iii) and (iv), we find that some functional forms of coupling for certain ranges of coupling strengths gives rise to nonMarkovianity. For example in case (iii), polynomial forms of coupling always give rise to nonMarkovian system dynamics, while exponential coupling give rise to nonMarkovian system dynamics only for certain ranges of parameter values. In case (iv) we find that a crossover from nonMarkovianity to Markovianity can be achieved by varying the strength of coupling. We also calculate, the extremal values of coupling parameter beyond which nonMarkovianity can no longer be detected. Thus we see, these extremal values act as critical values for transition from nonMarkovian to Markovian regime. It is worth mentioning here that, for the purpose of detecting nonMarkovianity we use RivasHuelgaPlenio (RHP) measure of nonMarkovianity as proposed in^{38}. Although there are different approaches of defining Markovianity and each approach represent different aspects of Markovianity, for the purpose of the present paper we choose, detection by the RHP measure as the definition of Markovianity.
Similar works on this line were done in^{39,40,41}. But in the first approach^{39}, the system qubit directly interacts with a single environment qubit and the rest of the environment qubits, only have an indirect effect on the system via the environment qubit directly attached. Also, the coupling parameters involved do not have any time dependence. In the second approach^{40}, the transition from Markovianity to nonMarkovianity was shown with a two tier environment; the first one being a multiplespin system, while the second one was a bosonic bath. Also in^{41}, the coupling between the system and individual environment qubits were constant in space and time. We take into account all these factors and present a detailed study of a spin environment and cover all the relevant cases.
In the background section, we discuss the relevant background required for following the techniques used in the paper. In the next section, we present our model, followed by a section, where we introduce different types of couplings and analyze them. Finally, we present the results of our analysis, before concluding in the last section.
Background
In this section we present the relevant background of Markovian dynamics and the definitions used in the paper. We also describe the measure of entanglement for twoqubit systems, which will also be used to quantify nonMarkovianity of our dynamics.
Quantum Markovian dynamics
A discrete time stochastic process is called Markovian (Markov chain) if the state of the system at time t_{n} depends solely on the state of the system at time t_{n−1}. This concept of Markov chain can be extended to the continuous time stochastic processes as well^{2}. However, generalizing it to quantum dynamics is a difficult task. Numerous prescriptions have been proposed to capture different aspects of quantum Markovianity. Broadly these prescriptions can be classified into two classes: information backflow^{42,43,44,45,46,47} and completely positive divisibility (CPdivisibility)^{1,48}.
Information backflow
The information backflow approach is inspired from the fact that a Markovian dynamics is characterized by unidirectional flow of information from the system to the environment. As for example, in the Lindblad master equation^{49}, the nonnegativity of the entropy production rate signifies unidirectional information flow from system to the environment, and thereby, is a signature of Markovianity. A dynamics is called Markovian from the information backflow approach, if some information quantifier decays over time in a monotonic way. Any departure from monotonicity of such quantifier is seen as a backflow of information from the environment, back to the system. Different quantifiers of information like distinguishability of states^{42}, measure of entanglement^{38}, quantum mutual information^{44}, etc has also been suggested for this purpose. Each quantifier provides a different definition of Markovianity; all of which, are not in general equivalent. Only recently, there has been attempts to unify all these different definitions^{45,46,47} to provide a unified approach to information backflow.
CPdivisibility
Any dynamical process, given by completelypositive (CP) trace preserving (TP) map Λ_{t} representing evolution up to time t is called CPdivisible if
where V_{t,s} is CP for any t ≥ s ≥ 0, and ° denotes composition.
Although the most general description of Markovian dynamics is given by CPDivisibility^{1,48}, for our purpose we consider information backflow, in terms of measure of entanglement i e. the RHP measure, as the description of Markovianity.
Detecting nonMarkovianity through Entanglement
Let us first discuss entanglement measure of a twoqubit state. The entanglement between two twolevel systems (two qubits) can be characterized by the PeresHorodecki criterion^{50,51} which states that a twoqubit state ρ_{as}, shared between a system qubit s and an ancilla qubit a, is entangled if and only if the partial transpose of this state, i.e. \({({\rho }_{{\rm{as}}})}^{{T}_{s}}\), is not a positivesemidefinite operator i.e. \({({\rho }_{as})}^{{T}_{s}}\,\ngeqq \,0\). Notably, for a twoqubit entangled state, the operator \({({\rho }_{{\rm{as}}})}^{{T}_{s}}\) has exactly one negative eigenvalue λ^{52,53}. Thus λ may be used as a measure of entanglement for the state ρ_{as}. Formally, the entanglement measure can be defined as follows
where, \({\Vert A\Vert }_{1}={\rm{Tr}}\sqrt{{A}^{\dagger }A}\) is the trace norm of a matrix A. Note that, E(ρ_{as}) is nothing but the negativity of the bipartite state ρ_{as}^{54}. We will use this measure of entanglement as the quantifier for ascertaining Markovianity of the dynamics from the information backflow approach. Using entanglement to detect nonMarkovianity was first used by Rivas, Huelga and Plenio in^{38}, and this measure has been so called the RHP measure of nonMarkovianity. Following their technique we attach an ancilla to the system, on which a dynamical map Λ_{t} is acting. Following the information backflow approach, the dynamical map Λ_{t} is called Markovian if \(E(({\mathbb{1}}\otimes {{\rm{\Lambda }}}_{t})[{{\rm{\Phi }}}^{+}\rangle \langle {{\rm{\Phi }}}^{+}])\) is a monotonically nonincreasing function of time t, where \({{\rm{\Phi }}}^{+}\rangle \langle {{\rm{\Phi }}}^{+}\) is the maximally entangled state, given by
The Model
In this section, we present our model and discuss the motivation behind choosing it. We also describe the technique in detail, in which nonMarkovianity in the system dynamics is detected. We consider two qubits, one of which is called the system (s) and the other, the ancilla (a). The system qubit is placed in an environment consisting of N noninteracting qubits (see Fig. 1). We take the interaction between the system qubit and the environment in the following form,
where \(0\rangle \) and \(1\rangle \), respectively represent the ground and excited states of each qubit.
The coupling strength \({\tilde{g}}_{n}(t)\) is a complex number which can be timedependent as well as sitedependent, and α is a real parameter with the dimension of frequency. The extra factor α is introduced to make the coupling strengths \({\tilde{g}}_{n}(t)\) dimensionless. For all practical purposes α can be assumed to be 1. The free Hamiltonians of the system and the environment are respectively given by,
It is convenient to work in the interaction picture where we replace the total Hamiltonian \(H={H}_{s}+{H}_{e}+\)\({\tilde{H}}_{{\rm{se}}}\equiv {H}_{0}+{\tilde{H}}_{{\rm{se}}}\) by the interaction picture Hamiltonian \({H}_{{\rm{se}}}(t)=\exp (i{H}_{0}t/\hslash ){\tilde{H}}_{{\rm{se}}}\,\exp (\,\,i{H}_{0}t/\hslash )\) which reads,
where δω_{n} = ω_{s} − ω_{n} and \({g}_{n}(t)={\tilde{g}}_{n}(t){e}^{i\delta {\omega }_{n}}\). Henceforth, our discussion will be based on the Hamiltonian H_{se}(t). We also consider the initial state of the environment to be in the thermal state,
where p = (1 + e^{−β})^{−1} and β is a positive real parameter which can be identified as the inverse of the temperature T of the environment.
Motivation behind the model
Here we argue that, our model is a close approximation to the ‘spin bath’ model^{34,35} at low temperatures. Note that the Hamiltonian in Eq. (7) can also be written as,
where \({\sigma }_{+}=0\rangle \,\langle 1\) and \({\sigma }_{}=1\rangle \langle 0\). When we compare Eq. (9) with the usual Hamiltonian of a spin bath model^{34,35,36} in the interaction picture, given by,
we find that the only difference comes from the 0〉 〈0 factors arising in Eq. (9), which are replaced by \({\mathbb{1}}\) for the spin bath Hamiltonian. As a result of this difference, the dynamics of the spin bath model is not entirely the same as our model. In the former, an exchange of one quanta of energy takes place between the system and individual environment qubit, when the rest of the environment qubits are allowed to be in any state, whereas in the later, the exchange will only take place when the rest of the environment qubits are in their ground state. This difference, although significant in general, will not play a major role when the state of the environment is close to the ground states, or in other words, temperature of the environment is low. Note that low temperature of environment correspond to values of p in Eq. (8), which are very close to 1, and this also confirms the fact that for low temperatures ρ_{e} is close to the ground state. Thus we see for low temperatures our model serves as a close approximation to the spin bath model. The main advantage of our model is the fact that our Hamiltonian is easily diagonalizable, and for certain types of couplings, as we discuss later in detail, allows for exact determination of the system dynamics in terms of Kraus operators, for any number of environment qubits.
We also stress that, although our model shows similarity to the spin bath model for low temperatures, we find solutions and analyze the dynamics of our model for any temperature whatsoever. The reason behind this is that our model being analytically solvable for certain types of couplings, allows for an opportunity to exactly solve the dynamics for any number of environment qubits, which is not often the case for systems with large number of spins. Note that, even for the spinbath Hamiltonian, it is not easy to find the exact solution for nonzero temperatures.
Diagonalizing the Hamiltonian of our model
There are only two nonzero eigenvalues of the Hamiltonian H_{se}(t) and they are,
corresponding to the eigenvectors,
where \({\xi (t)\rangle }_{se}={0\rangle }_{s}\otimes {{\beta }_{0}\rangle }_{e}\) and,
Thus, the time evolution operator U(t, 0) corresponding to the Hamiltonian H_{se} is,
where \({\mathscr{T}}\) represents time ordering.
The ancilla qubit is used as a probe to characterize the nonMarkovianity of the dynamics of the system in the presence of the environment. In order to do so we prepare the system and ancilla qubits in a maximally entangled state \({{\rm{\Phi }}}^{+}\rangle \), as given in Eq. (3). Due to the interaction of the system qubit with the environment, the entanglement between the system and the ancilla qubit will evolve with time. The deviation of this time evolution of the entanglement, from monotonic decay is used to establish the nonMarkovian character of the dynamics. Note here, that this idea was used by Rivas et al.^{38} to devise a measure of nonMarkovianity. In the present paper, we follow this technique to consider the system dynamics to be nonMarkovian whenever the entanglement between system and ancilla, as described above, shows nonmonotonic behaviour, otherwise we consider the dynamics to be Markovian.
The joint initial state of the system plus ancilla plus environment is of the form,
which evolves to,
Therefore, reduced timeevolved systemancilla state can be calculated by tracing out the environment part,
System  Environment Couplings
In this section, we introduce various classes of systemenvironment coupling, and in each case, we study their effect on the evolution of the systemancilla joint state. We classify all the couplings into four major classes: (A) when the coupling parameter g_{n}(t) is independent of the site index n (homogeneous) and timeindependent; (B) when g_{n}(t) is inhomogeneous but timeindependent; (C) when g_{n}(t) is homogeneous but timedependent, and (D) when g_{n}(t) is inhomogeneous and timedependent. For each class, we calculate the entanglement of the time evolved state of systemancilla, and thereby try to characterize the nonMarkovian behaviour of the system dynamics. Henceforth, we assume α to be 1.
Case A: Homogeneous and timeindependent coupling
We have here the simplest situation, where the coupling of the system with all the environment qubits are uniform and timeindependent i.e. g_{n}(t) = g, a constant. As a result, the nonzero eigenvalues of the Hamiltonian, as given in Eq. (11), takes the form \({\varepsilon }_{\pm }(t)=\pm \,\varepsilon =\)\(\pm \,\hslash \sqrt{N}\mathrm{\ }g\equiv \hslash {\omega }_{0}\), where \({\omega }_{0}=\sqrt{N}\mathrm{\ }g\) is a constant with the dimension of frequency. The timeevolution operator U(t, 0) is of the form,
Using the above form and the form of ρ_{e} given in Eq. (8), we find the Kraus operators K_{mn}(t) of system dynamics, which are defined in the following way,
where the N^{2} Kraus operators are given by,
where m, n = 1, …, N, log x refers to \({\mathrm{log}}_{2}\,x\), and s_{n} is the number of 1’s in the binary equivalent of n. For example, if n = 6, then the binary equivalent of n is 110. Therefore s_{n} = 2.
We then find time evolved state of the systemancilla, using Eqs (15–17),
where \({\kappa }_{0}={p}^{N1}{\sin }^{2}({\omega }_{0}t)\) and \({\delta }_{0}=2{p}^{N1}{\sin }^{2}(\frac{{\omega }_{0}t}{2})\). The only possible negative eigenvalue of \({[{\rho }_{{\rm{as}}}(t)]}^{{T}_{s}}\), if any, is of the form,
We present the plot of E(ρ_{a}s(t)) = λ(t) versus time, later in the Result section.
Case B: Inhomogeneous and timeindependent coupling
Consider a system, where a single twolevel system (perhaps an ion as an impurity) is placed in a spin lattice. The lattice sites, closest to the impurity interacts very strongly with the system, while, as we go away from the impurity site, the strength of interaction becomes weaker and weaker. In such cases, we have a scenario similar to our model, and the interaction parameter g_{n}(t) is inhomogeneous, but there is no explicit time dependence. Therefore, g_{n}(t) = g_{n}. Hence, ε_{±}(t) = ±ε = ℏω in Eq. (11) are also timeindependent. Note, in this case also \(\omega =\sqrt{{\sum }_{n\mathrm{=1}}^{N}{g}_{n}{}^{2}}\) is a constant with dimensions of frequency. Following this, the analysis is same as in the last subsection. As a result, the evolution operator U(t, 0), the time evolved state ρ_{as}(t) and the only possible negative eigenvalue, if any, of \({[{\rho }_{{\rm{as}}}(t)]}^{{T}_{s}}\) for this case are of the same forms as in Eqs (18, 20–22) respectively, except for ω_{0}, in appropriate places, replaced by ω.
Case C: Homogeneous and timedependent coupling
So far we have considered only couplings which are independent of time. In this section, we consider timedependent and homogeneous couplings. We take an arbitrary real function of time, which is independent of site index n i.e. g_{n}(t) = g(t). Note that our coupling operator between system and individual environment qubit, as given in Eq. (7), is of the form σ_{+} ⊗ σ_{−} + σ_{−} ⊗ σ_{+} which can also be expressed as σ_{x} ⊗ σ_{x} + σ_{y} ⊗ σ_{y}. Thus, our systemenvironment coupling is a special case of the XY coupled Hamiltonian. Such coupling with timedependent coefficients have been used to show nontrivial entanglement dynamics^{55,56}.
Fortunately, the Hamiltonians H_{se}(t) in this case commutes at different times, which makes the analysis similar to the one in case B. The only difference being the nonzero eigenvalues, in Eq. (11), to be of the form \({\varepsilon }_{\pm }(t)=\pm \,\hslash \sqrt{N}g(t)=\pm \,\varepsilon (t)\), which is no longer constant in time. The whole treatment of the dynamics of the system and the ancilla remains the same if we replace ω_{0} and ω_{0}t, in Eqs (18, 20–22), by \(\sqrt{N}\mathrm{\ }g(t)\) and Ω(t), respectively, where,
Case D: Inhomogeneous and timedependent coupling
The most general class of coupling g_{n}(t) is when it depends on both the site n and time t. The interaction Hamiltonian in such a situation does not commute at different times and this makes the calculation for solving the dynamics difficult. However, we can use numerical methods to simulate the timeevolution and get the solution for ρ_{as}(t). One can obtain the following results analytically before starting the simulation part.
Analytical Part
Two of the eigenvalues of the Hamiltonian in Eq. (7) are nonzero, as given in Eq. (11). The remaining (2^{N+1} − 2) of the eigenvalues are zero. A possible choice for these null space eigenvectors are found in the following way:
Step I: We first feed the eigenvectors (corresponding to nonzero eigenvalues) given in Eq. (12) as rows of a matrix A. Note, A is a 2 × 2^{(N+1)} matrix
Step II: By row reduction method^{57}, we find out a basis \( {\mathcal B} \) for the Null space of A. Note that \( {\mathcal B} \) is not necessarily orthonormal.
Simulation Part
Obtaining an orthonormal basis \( {\mathcal B} ^{\prime} \) from \( {\mathcal B} \) analytically is a challenging job. We, therefore resort to numerical techniques for this case.
Step I: From \( {\mathcal B} \), using GramSchmidt Orthonormalization procedure^{57}, we find an orthonormal basis \( {\mathcal B} ^{\prime} \). Note, \( {\mathcal B} ^{\prime} \) forms the set of eigenvectors of the Hamiltonian, corresponding to zero eigenvalues.
Step II: As, the eigenvectors are timedependent, the Hamiltonian is not differenttime commuting. Hence, the evolution operator may be found numerically from the following expression,
where \({\mathscr{T}}\) represents time ordering.
Step III: We evolve the initial ancilla systemenvironment state ρ_{ase}(0) by the unitary operator \({U}_{ase}(t,\,\mathrm{0)}={{\mathbb{I}}}_{a}\otimes {U}_{se}(t,\,\mathrm{0)}\) and get the time evolved state ρ_{ase}(t).
Step IV: We trace out the environment from ρ_{ase}(t) and get ρ_{as}(t) = Tr_{e}[ρ_{ase}(t)]. We then evaluate our entanglement measure E(t) given in Eq. (2), on ρ_{as}(t) and plot it as a function of time.
Results
In this section, we show that some of the classes of the couplings that we have considered, always results in nonMarkovian dynamics. However, there are also some classes for which we can tune the parameters and find a transition from nonMarkovian to Markovian dynamics. In order to do so, we plot the entanglement dynamics between the system and ancilla, for each class, as a function of time and observed if there is any departure from monotonicity with time. As mentioned in Sec. III, this technique helps in characterizing nonMarkovianity present in the system dynamics. In Figs 2 and 3, we present entanglement dynamics for different classes of the systemenvironment coupling, considered in the previous section. Also in Table 1, we provide a concise summary of all the resuts obtained in this section. We now present our findings for each class of systemenvironment coupling.
Cases A and B
In Fig. 2(a,b), we plot the entanglement as a function of time for the homogeneous timeindependent, and the inhomogeneous timeindependent couplings, respectively, i.e, g_{n}(t) = g and g_{n}(t) = g_{n}, respectively. Note here, that g and g_{n}, for all values of n, are arbitrary complex functions. Analytic calculations for the entanglement measure, as given in Eq. (22) suggests a periodic behaviour for both the classes, which can also be seen in Fig. 2(a,b). As a result, we conclude in both of these classes of couplings, the dynamics is always nonMarkovian.
Case C
We consider homogeneous and timedependent couplings and find that if g(t) is some polynomial function of t, we will get Ω(t) as a polynomial function of t. This gives rise to a periodic function λ(t). As a result, the dynamics is nonMarkovian, in general. Interestingly, it was recently pointed out^{58}, that the Hamiltonian dialation obtained from the Choi state of the system dynamics, diverges whenever the system dynamics is timeindependent Markovian. Although, in our case, the systemenvironment Hamiltonian (equivalent to the dialated Hamiltonian)is not related to the Choi state of the system dynamics in the same way as in^{58}. As a result the conclusion of^{58} differs from ours. If g(t) = exp(−γt) then nonMarkovianity can be witnessed if the real part γ_{r} of γ fails to be positive or violates the inequality \(\alpha \sqrt{N}\ge {\gamma }_{r}\pi \). Figure 2(c,d) show the entanglement vs time plot for two values of γ_{r}; one of which violates the above mentioned inequality. We also consider the case \(g(t)=\frac{1}{1+\gamma t}\), and show in Fig. 2(e,f) that the dynamics is nonMarkovian for various values of γ. N and p.
Case D
For inhomogeneous timedependent couplings, the dynamics can be made both nonMarkovian and Markovian by choosing the strength of the coupling appropriately. We consider two special cases of inhomogeneous timedependent coupling: (i) \({g}_{n}(t)={e}^{{\gamma }_{1}nt}\), and (ii) \({g}_{n}(t)=\frac{1}{1+{t}^{n\gamma }}\). For simplicity, we have assumed α = 1. For analyzing coupling (i), we plot the systemancilla entanglement measure as a function of time in Fig. 3(a–d), for different values of the coupling parameter γ_{1}, at a fixed values of N and p. In Fig. 3(a–d), monotonically decreasing entanglement values show signs of Markovianity and nonmonotonic decay are evidence of nonMarkovianity. As expected, increasing the coupling parameter γ_{1} i.e., decreasing coupling strength, leads to the transition from nonMarkovian to Markovian dynamics. The figures also show an interesting feature that, after sufficient time, the entanglement in the systemancilla state saturates to fixed values irrespective of their Markovian or nonMarkovian nature. This feature can be signs of possible equilibration of the system ancilla state. Next, we find the extremal values of γ_{1} for which nonMarkovianity is witnessed. These extremal values serve as transition parameters from nonMarkovianity to Markovianity. On plotting these transition values as a function of N (see Fig. 3(i)), it appears that a saturation is reached as N is increased for values p = 0.5 and p = 1.0. We perceive, this is the result of the fact that for this type of coupling i.e. \({g}_{n}(t)={e}^{{\gamma }_{1}nt}\), the larger is the value of N, the smaller is its effect on the system dynamics. Although a definitive conclusion about whether the saturation persists over large N can only be made after computing the transition values for larger values of N, it is a computationally demanding process with the computational facilities available at our disposal. For coupling (ii), the dynamics shows nonMarkovianity for various values of γ, N and p, as shown in Fig. 3(e–h).
Conclusion
In this paper, we have addressed the question of how nonMarkovianity of a dynamics changes with the interaction between the system and the environment and also with size of the environment. We have taken a simple model constituting of a few qubits, which can also be seen as a close approximation to the spin bath model, for low temperatures. Even in this minimalistic scenario, we were able to find a transition from nonMarkovian to Markovian dynamics by tuning the systemenvironment interaction. This is somewhat counterintuitive as it is generally conceived that for having Markovian dynamics the bath/environment should have infinite degrees of freedom, although there are exceptions^{59}. We also found, in our model, that if the interaction Hamiltonian is timeindependent, the dynamics is always nonMarkovian, irrespective of the size of environment. Note that in this scenario for a general interaction Hamiltonian in the weak coupling limit, we generally get to see Markovian dynamics only for a very large size of bath; at least in the case of harmonic oscillator bath. The present scenario is different as we have considered spin environment and the systemenvironment interaction is of very specific type. In the case of siteindependent interaction, polynomial forms and certain cases of exponential forms of interaction show nonMarkovianity. Lastly, we study timedependent and sitedependent interaction for certain forms of systemenvironment coupling. In this last case, we also saw a transition from nonMarkovian to Markovian regime. Interestingly, the transition values appear to saturate to a certain value depending on the initial temperature of the environment, as the number of environment qubits increases. Examining this type of spin environment is recently drawing some amount of interest^{33}. Studies on similar lines was also done recently in^{36}, where an analysis of a qubit system interacting with a sea of spins was given. A number of questions arise from the present study: whether such a transition can be found by considering more general forms of interaction, what happens if along with systemenvironment interaction there is some interaction present among the environment particles themselves, etc. One may also find it useful to check, whether the aforesaid saturation of transition parameters is a general feature of interaction that exhibits nonMarkovian to Markovian transition.
Change history
06 December 2019
An amendment to this paper has been published and can be accessed via a link at the top of the paper.
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Acknowledgements
The authors would like to thank R. Lo Franco, C. GonzlezGutirrez, F. Plastina and T.J.G. Apollaro for providing useful inputs on the manuscript. SC, AM, and SG gratefully acknowledge discussions with Samyadev Bhattacharya and Chiranjib Mukhopadhyay.
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A.M. designed the study. S.C., A.M. and D.M. carried out analytical derivations, numerical analysis and prepared the manuscript under the guidance of S.G and S.K.G.
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Chakraborty, S., Mallick, A., Mandal, D. et al. NonMarkovianity of qubit evolution under the action of spin environment. Sci Rep 9, 2987 (2019). https://doi.org/10.1038/s41598019391402
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DOI: https://doi.org/10.1038/s41598019391402
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