Memory-keeping effects and forgetfulness in the dynamics of a qubit coupled to a spin chain
Using recently proposed measures for non-Markovianity [H. P. Breuer, E. M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)], we study the dynamics of a qubit coupled to a spin environment via an energy-exchange mechanism. We show the existence of a point, in the parameter space of the system, where the qubit dynamics is effectively Markovian and that such a point separates two regions with completely different dynamical behaviors. Indeed, our study demonstrates that the qubit evolution can in principle be tuned from a perfectly forgetful one to a deep non-Markovian regime where the qubit is strongly affected by the dynamical back-action of the environmental spins. By means of a theoretical quantum process tomography analysis, we provide a complete and intuitive characterization of the qubit channel.
pacs:03.65.Yz, 75.10.Pq , 42.50.Lc
In the study of open quantum systems, quite often the Markovian approximation has been a useful starting point to describe their dynamics. Only very recently, new and powerful tools have been designed in order to tackle the important question of explicitly quantifying the non-Markovian character of a system-environment interaction or a dynamical map breuer09 (); rivas09 (); wolf08 (). Such a task is extremely important, given that non-Markovian effects are known or expected to occur in a wide range of physical situations, especially in the realm of solid-state devices where a system of interest is often exposed to memory-preserving environmental mechanisms. As a specific example, switching impurities have been shown to affect superconducting devices in various regimes martinis ().
From a fundamental point of view, devising a reliable way to actually quantify non-Markovianity is very useful in light of the plethora of frequently ad hoc or technically rather involved approaches put forward so far in order to study memory-keeping environmental actions. A few instances emerge as promising measures for the non-Markovian nature. Wolf et al. wolf08 () have proposed to quantify the degree of non-Markovianity of a map by considering the minimum amount of noise required in order to make the evolution of a system fully Markovian. On the other hand, in Ref. rivas09 (), Rivas and co-investigators discussed two approaches, founded on the deviations of a given map from full divisibility. Finally, Breuer et al. proposed a way to quantify non-Markovian effects by looking at the back-action induced on the system under scrutiny by its memory-keeping environment breuer09 (). Further approaches have been considered and measures have been proposed which are based on the use of other interesting instruments such as the quantum Fisher information sun09 ().
In this paper, we focus on the non-Markovianity measure proposed in Ref. breuer09 () to study the dynamics of a qubit coupled to a spin environment described by an model in a transverse magnetic field. Our aim is to analyze a simple and yet non-trivial system-environment set, displaying a broad range of behaviors in the parameter space, in order to relate the features of the non-Markovianity measure to the known spectral and dynamical structures of the spin system. The choice of the model has been also dictated by the versatility demonstrated by spin-network systems in the engineering of protocols for short-range communication boseetal () and also for the investigation of the interplay between quantum-statistical and quantum-information-related aspects rmp ().
Intuitively, one would expect that the coupling to a spin environment always leads to a non-Markovian dynamics for the qubit. This can be justified by noticing that the environmental correlation time is non-zero even in the weak coupling regime and by conjecturing that, by increasing the coupling strength, a Markovian approximation can only become less valid. However, we demonstrate that this is not at all the case and that peculiar behaviors occur at intermediate couplings. The chosen measure for non-Markovianity, in fact, turns out to be identically null at specific dynamical regimes, thus demonstrating the absence of a net re-flux of information from the environment back to the qubit. Our study of non-Markovianity is thus novel and makes the use of measures for its quantification a valuable tool for the identification of special conditions in the parameter space of the environment. Moreover, our study opens up the possibility to exploit the different memory-keeping regimes induced by controlling and tuning the properties of the environment to effectively “drive” the qubit dynamics in a non-trivial and potentially very interesting way. The effective evolution of our two-level system could be guided across different regimes, ranging from strong environmental back-action to completely forgetful dynamics typical of a Markovian map. The potential of such flexibility for reliable control at the quantum level is a topic that will be explored in future.
The reminder of this paper is organized as follows. In Sec. I we introduce the model under scrutiny and revise the basic principles behind the chosen measure for non-Markovianity. We study a few cases amenable to a full analytical solution and highlight how, in a few of such instances, a simple experimental protocol can be designed for the inference of the properties of the qubit dynamics. Most importantly, we reveal the existence of an operating regime where the chosen measure is strictly null and relate such an effect to intriguing modifications occurring at the level of the energy spectrum of the qubit-environment system. Sec. II is devoted to the formal characterization of such peculiar point in the parameter space. We first demonstrate divisibility of the corresponding dynamical map, hence its Markovian nature, and then perform a theoretical analysis based on the use of quantum process tomography to quantitatively infer its properties. Finally, in Sec. III we draw our conclusions.
I The model and the measure
We consider a qubit coupled to a chain of interacting spin- particles. The qubit is described by the spin- vector operator , while the operator () corresponds to the spin located at site of the chain . The logical basis for the spins and is given by with . The Hamiltonian ruling the intra-chain interaction is taken to be of the -Heisenberg type (we set )
where is the local field applied at site . is open-ended with ’s and ’s being not necessarily uniform. The qubit is coupled to the first spin of the environment, embodied by , via an exchange interaction of strengths and is subjected to a local field according to
In order to determine the time-evolution of , we resort to the Heisenberg picture and the formal apparatus put forward in Refs. DiFrancoEtal07 (), which provides particularly powerful tools for the study of the many-body problem embodied by . Using the operator-expansion theorem and the algebra satisfied by Pauli matrices, we find that the time evolution of the components of reads
where () are the Pauli operators for the spin at site and . The time-dependent coefficients and are the components of the -dimensional vectors and defined by
where stands for transposition, the vector has components and the tri-diagonal adjacency matrix has elements
Notice that we have labelled columns and rows of matrices and -dimensional vectors using indices ranging from to . The coefficients and are obtained from Eqs. (4) and (5) by replacing with . Both and can be easily diagonalized by orthogonal matrices and such that and , with a diagonal matrix whose elements are the (positive) square roots of the eigenvalues of . Consequently, Eqs. (4) and (5) can be fully summed up to give
where and are diagonal matrices with elements and .
By using Eqs. (3), one can determine the time evolution of the state of as . In the evaluation of the expectation values required to determine we assume that and are initially uncorrelated. The conservation rule and the property imply
In order to evaluate the above equations one needs multi-spin correlation functions, involving, in particular, the degrees of freedom of . To this end, we consider it to be in its ground state in what follows. However, as it will be shown later on, the value of the measure of non-Markovianity chosen for this work is independent of the state of , provided that holds, as it does for the ground state with no broken symmetry.
By using Eqs. (8), one can finally determine . The interaction with the spin chain acts for as a dynamical map such that . The properties of the map depend on the relative weight of the various parameters entering . Our aim is to characterize the non-Markovian nature of as a function of these parameters.
To pursue our task, we consider the measure proposed in Ref. breuer09 (), based on the study of the time-behavior of the trace distance between two single-qubit density matrices . The trace distance is such that when the two probed states are completely distinguishable, while it gives for identical states NC (). The degree of non-Markovianity of the dynamical map , is defined as
where the maximization is performed over the states and is the time window such that
The function , which has been dubbed flux of information in Ref. breuer09 (), encompasses per se the condition for revealing non-Markovianity of an evolution: the mere existence of even a single region where is sufficient to guarantee the non-Markovian nature. Conceptually, in fact, accounts for all the temporal regions where the distance between two arbitrary input states increases, thus witnessing a re-flux of information from the environment to the system under scrutiny. Such re-flux of information amplifies the difference between two arbitrarily picked input states evolved up to the same instant of time. A Markovian dynamics is such that the above-mentioned re-flux never occurs and always. For the case at hand and for two generic input density matrices , we find , where
with where and . We have used the notation with and . It is worth noticing that in the above equation the initial state of the environment is completely absent, so that the environment’s multi-spin correlators are not relevant. Eq. (11) can be recast into a much more intuitive form by referring to the Bloch vectors representative of the state (we have chosen as the mean value of the spin- operator rather than that of the more usual Pauli operator to avoid the appearance of irrelevant factors). By calling the difference between the vectors of two input states, we have
One can thus write the flux of information between the qubit and the environmental chain as
where , , and . Due to the non-negativity of , the condition for non-Markovian dynamics can be simply stated as . Numerically, it turns out that the maximum in the corresponding measure of non-Markovianity is achieved for being antipodal pure states lying on the equatorial plane of ’s Bloch sphere (we will come back to this point later in this paper). This considerably simplifies the necessary condition for memory-keeping dynamics to the form .
Although our analysis can be carried out without major complications in the general case, in order to simplify the presentation, from now on we restrict ourselves to the case of a uniform spin environment with equal isotropic couplings between every pairs of nearest neighboring spins (-model) and set this coupling constant as our energy (and inverse time) unit. We thus consider , a condition under which the measure of non-Markovianity becomes
where . The corresponding rate of change of the trace distance is
As , the sign of is determined by , regardless of the pair of input density matrices . Yet, we should look for the states maximizing the contributions to the trace distance within these time intervals. As , such optimization is achieved for and ; that is, antipodal pure states on the equatorial plane of the Bloch sphere, in line with the (more general) numerical findings reported above. The condition for non-Markovianity can be further elaborated as .
To pursue the task of evaluating in the quite rich parameter space of our model, we start by considering the case of a qubit resonant with the spin environment, and assess the special case of first. It can be shown analytically that in this case one gets and ApollaroEtal10 (), where is the Bessel function of order and argument . Correspondingly, and . For those states with and which maximize , we have the flux
with being the sign function. Eq. (16) is independent of , which is joint result of the condition and the invariance of the trace distance under the global unitary transformation embodied by the operator needed in order to pass to the interaction picture. The time windows where are determined by the chain-rule of the zeros of the Bessel functions. Overall, we get that the flux is positive for a time where () is the zero of the Bessel function of order 1 (2). From this special case and the general considerations reported above, we learn that only depends on the detuning .
Due to this fact, the simple case in which all of the magnetic fields are absent (), which can be tested experimentally in an easy way, allows to draw some interesting conclusions on the more general case . In absence of magnetic fields, indeed, the behavior of and can be extracted directly by monitoring the dynamics of the qubit. To show that this is indeed the case we start by noticing that, if , we have ApollaroEtal10 (), so that both the trace distance and the flux of information are determined by , which represents the squared length of the Bloch vector, . This implies that, in order for to experience a re-flux of information from , has to be a strictly non-monotonic function. Therefore, we observe non-Markovian dynamics when the Bloch vector of the qubit is alternatively shrunk and elongated during its evolution. This can be witnessed by reconstructing the density matrix of using standard quantum state tomography techniques statetomo (); NC (), which are routinely implemented in a variety of physical setups. However, there is also an interesting alternative which does not require full state reconstruction. From the first of Eqs. (8), we get that . Thus, in the spirit of the proposals put forward in Refs. DiFrancoEtal07 (); DiFrancoEtal08 (), by preparing the state of in one of the eigenstates of [a choice that would be perfectly consistent with our results on the input states to be used for the calculation of ] we can measure to determine the non-Markovianity of the qubit dynamics, which would be revealed by its non-monotonic time behavior.
Other points in parameter space exist for which the model is amenable to an exact analytic solution. However, in order to give a complete overview of the behavior of , we first resort to numerical techniques to solve our model. The results of such an analysis are shown in Fig. 1, where the quantitative degree of non-Markovianity is shown as a function of (we set ) in the isotropic case and for equal intra-chain and qubit-environment coupling strengths.
A highly non-trivial behavior followed by is revealed. The largest deviation from a Markovian dynamics is achieved at while, for , decreases monotonically to zero. By further increasing we see that achieves a very broad maximum around the saturation point of the environmental chain. Finally, it goes to zero for , as it should be expected given that this situation corresponds to an effective decoupling of the qubit from the environment ApollaroEtal10 (). By generalizing our study to the case of , we emphasize the presence of a Markovianity point at . It turns out that this point separates two regions in which the dynamics of the qubit (although being non-Markovian in both cases) is completely different. For , indeed, the qubit tends toward a unique equilibrium state at long times, irrespective of the initial condition; that is, the trace distance goes to zero after some oscillations. For larger detunings, on the other hand, the trace distance does not decay to zero, implying that some information about the initial state (and in particular, about the relative phase between its two components) is trapped in the qubit.
The Markovianity point at (with ) is one of those points in parameter space for which the model can be treated fully analytically yueh04 (). We have and ApollaroEtal10 (), so and . As a consequence, the corresponding non-Markovianity measure is always zero at this point of the parameter space, which is a very interesting result due to the relatively large coupling strength. Such a feature, in fact, would intuitively lead to exclude any possibility of a forgetful dynamics undergone by the qubit. Yet, this is not the case and a fully Markovian evolution is in order under these working conditions. A deeper characterization of this Markovian dynamical map is given in Sec. II.
As represents the energy scale of the qubit-environment interaction, it is natural to expect that significant changes in occur as this parameter varies. In particular, we find that i) a Markovianity point with only exists for and ii) tends progressively towards a monotonically decreasing function of if grows from to [see Fig. 2]. Strikingly, at and , the adopted measure of non-Markovianity diverges, as it can be checked by using the analytic integrability of the qubit-chain interaction at this point in the parameter space. Indeed, at , we have and . Integrating over all the positive time intervals determined by means of the usual chain rule we get , thus witnessing a strong back-action of on the state of the qubit. The divergence of should not surprise as it is common to other situations with spin-environments, such as the so-called central-spin model where a single qubit is simultaneously coupled to independent environmental spins via Ising-like interactions (see Breuer et al. in Ref. breuer09 ()). We provide a physical explanation for the enhanced non-Markovian nature of the qubit dynamics simply by looking at the spectrum of the Hamiltonian ruling the evolution of the qubit-chain system. For , the spectrum of exhibits a continuous spectrum (a band of extended eigenstates) that is lower- and upper-bounded by two discrete energy levels whose eigenstates are localized at the sites occupied by the qubit and the first spin of nota (). As a consequence, a certain amount of information remains trapped into such a localized state, bouncing back and forth between the qubit and the first spin and therefore mimicking a highly non-Markovian dynamics characterized by strong back-action, so that diverges.
An analysis similar to the one performed just above allows us to obtain an intuition for the behavior of the measure of non-Markovianity near this point. For the spectrum of the system shows one eigenenergy out of the band and its corresponding eigenvector is localized around the site occupied by . This can explain the information trapping that occurs for . For definiteness, in what follows we report explicit results for the case . Therefore, from now on, we consider .
In order to determine the existence of a more general connection between the emergence of localized eigenstates in the system-environment spectrum and a point of zero- in the qubit evolution, we analyze the -plane to find out where localized eigenstates appear and then evaluate the corresponding degree of non-Markovianity. In doing this, we take advantage of the fact that the Hamiltonian describing an environmental -model has the same single-particle energy spectrum as a tight-binding model with an impurity. Following the approach given in Ref. pury91 (), we deduce that, in the plane, the parabolae define regions with respectively zero, one and two localized energy levels out of a continuous-energy band (see Fig. 3).
The central region with no localized energy state and the zones with only one localized state (either a upper-lying or lower-lying one with respect to the continuous band) correspond to , although finite. The frontiers of such regions, marked by the parabolae, give the limiting values of for the appearance of the localized states and are such that only for , whereas for the measure of non-Markovianity is non-null and finite. Finally, the regions with two localized states and their frontier with the no-localized state region have , in line with the analytical results discussed above.
In particular, we remark once again that the Markovianity points (when they exist, i.e. for ) stay on the parabolae; that is, they are found to occur at the onset for the existence of one discrete eigenstate outside the energy band. This kind of state contains a spatially localized spin excitation with a localization length that decreases with increasing nota (). Precisely at the border, the localization length becomes as large as the length of the chain itself, so that all of the environmental spins are involved in (or share excitation of) the initial state. This can intuitively justify the fact that is zero there. For , on the other hand, the first spin of the environment becomes more important for the dynamics of the qubit (which is “strongly coupled” to it) and an information exchange is always found to occur between them, irrespectively of the existence of the discrete level. This information re-flux becomes more and more pronounced with increasing and decreasing .
To stress once more the close relationship between the non-Markovianity measure and the properties of the overall Hamiltonian , we report here the energy distribution of the excitations which are present in the initial state of the system (given by the product of an equatorial state for the qubit times the ground state of the environment). These excitations are spin-less fermions of the Jordan-Wigner type and the procedure to obtain them is the one described, e.g., in Ref. apollaro08 (). Fig. 4 reports the average value of the excitation number in the initial state vs single-particle-energy for various values of the magnetic field. The plot shows what happens near a Markovianity point: the energy distribution of initial-state excitations becomes flat (i.e., structure-less) at the point, while it shows a maximum for lower values of , corresponding to a finite value of , and a spike for , corresponding to the discrete level giving rise to information trapping.
From this discussion we conclude that the measure of non-Markovianity of the qubit dynamics is in fact a detector of general aspects of the full qubit+environment system and that various features of can be related to general characteristics of the overall many-body problem described by the full Hamiltonian model.
Ii Characterization of the point of zero-measure
The occurrence of a null value of at and deserves a special attention. Naively, one could expect a special behavior to occur at the chain saturation point (i.e. at ), where the intrinsic properties of the environmental system are markedly different from the situation at . To the best of our knowledge, indeed, no significant dynamical feature has been reported for the model under scrutiny away from saturation.
ii.1 Characterization of the dynamical map: formal features and divisibility
The aim of this Section is to characterize the dynamical map that we obtained for the qubit under these conditions.
The dynamical map transforms the elements of the input density matrix as
with , and , and where has already been defined in Sec. I. We introduced here the function , written in terms of the magnetization and the two-point longitudinal correlation function
with , , and being the Fermi wave number, see Ref. wonmin ().
The condition for divisibility stated in Ref. breuer09 () implies the existence of a completely positive dynamical map such that, for two arbitrary instants of time and , we have . Here is the dynamical map in Eq. (19). Any dynamical map that is divisible according to the above definition is Markovian. This implies that non-divisibility is a necessary condition for memory-keeping effects in the evolution of a system. A dynamical connection between the states and can be straightforwardly found to be given by the map changing the elements of the qubit state at time into
Therefore, in order to ensure the divisibility of , we should investigate the complete positivity of . To this purpose we make use of the Choi-Jamiolkowski isomorphism CJ () and prove the complete positivity of by checking the non-negativity of , where is the density matrix of one of the Bell states ruskai () and is the identity map. By choosing , the action of the map determines the following non-zero matrix elements (up to an irrelevant factor ) at time
Here, we have used with the primed (unprimed) indeces corresponding to the evolving (non-evolving) qubit. The condition for positivity of the composite two-qubit map turns out to be equivalent to the positivity condition of the single-qubit one given in Eqs. (21), which is in turn translated into the inequality with
In Fig. 5 we show the typical behavior of at the Markovianity point [panel (a)] and away from it [panel (b)]. We have taken the qubit as prepared in , which is a significant case as equatorial states in the Bloch sphere are those optimizing the calculation of . Although this is simply a representative case, we have checked that for a uniform distribution of random initial states of , no significant quantitative deviations from the picture drawn here are observed. Clearly, by moving away from , temporal regions where are achieved. This demonstrates, from a slightly different perspective, the flexibility of the effective qubit evolution: a wide range of dynamical situations is spanned, from fully forgetful to deeply non-Markovian dynamics, strongly affected by the environmental back-action. The kind of evolution of can be determined by tuning the parameters of the environment and its interaction with it.
ii.2 Formal characterization of the channel through theoretical quantum process tomography
We now turn our attention towards the formal characterization of the channel achieved at the Markovianity point, so as to qualitatively explain the reasons behind the nature of the corresponding qubit evolution. We stress that this sort of investigation is meaningful only at this specific point in parameter space, where complete positivity is guaranteed. In principle, full information on the reduced dynamics of could be gathered from the Kraus operators such that with the density matrix of the generic qubit and . As their direct computation is not possible due to the complications of the coupling, here we gain useful information on the structure of the ’s by means of the formal apparatus for quantum process tomography NC (), that we briefly remind here.
The characterization of a dynamical map reduces to the determination of a complete set of orthogonal operators over which one can perform the decomposition so as to get
where the channel matrix has been introduced. This is a pragmatically very useful result as it shows that it is sufficient to consider a fixed set of operators, whose knowledge is enough to characterize a channel through the matrix . The action of over a generic element of a basis in the space of the matrices (with ) can be determined from a knowledge of the map on the fixed set of states and as follows
Therefore, each (with ) can be found completely via state tomography of just four fixed states. Clearly, as form a basis. From the above discussion we have
where we have defined so that we can write