Dynamical decoupling efficiency versus quantum non-Markovianity

We investigate the relationship between non-Markovianity and the effectiveness of a dynamical decoupling protocol for qubits undergoing pure dephasing. We consider an exact model in which dephasing arises due to a bosonic environment with a spectral density of the Ohmic class. This is parametrised by an Ohmicity parameter by changing which we can model both Markovian and non-Markovian environments. Interestingly, we find that engineering a non-Markovian environment is detrimental to the efficiency of the dynamical decoupling scheme, leading to a worse coherence preservation. We show that each dynamical decoupling pulse reverses the flow of quantum information and, on this basis, we investigate the connection between dynamical decoupling efficiency and the reservoir spectral density. Finally, in the spirit of reservoir engineering, we investigate the optimum system-reservoir parameters for achieving maximum stationary coherences.


I. INTRODUCTION
Dynamical decoupling (DD) techniques for open quantum systems are among the most successful methods to suppress decoherence in qubit systems [1,2]. The development and growth of technologies to achieve unprecedented control of individual quantum systems have paved the way to intense theoretical and experimental investigation of different DD schemes. Sophisticated control design have superseded earlier schemes such as the so-called "bang bang" periodic dynamical decoupling (PDD) [2] and its time-symmetrized version [3,4]. On the one hand concatenated DD schemes (CDD) have been developed to counter decoherence for general noise scenarios [5], on the other hand optimal approaches to minimize errors in specific noise settings have been discovered [6]. In both cases, a high sensitivity of the efficiency of the protocols to the pulse timing has been demonstrated.
The performance of all DD schemes crucially depends on the spectral properties of the noise causing decoherence and introducing errors. In Ref. [7], an exactly solvable pure-dephasing model was used to compare the efficiency of certain DD protocols in Ohmic, sub-Ohmic and super-Ohmic environments. This is important because of the increasing ability to engineering experimentally environmental properties, such as the spectral distribution [8].
It has been recently demonstrated [9,10] that, in pure dephasing models, the form of the environmental spectrum is linked to the concept of non-Markovianity in terms of information flow, as defined in Ref. [11]. According to this characterisation of open quantum system dynamics, memory effects arising from non-negligible correlations between system and environment give rise to a back flow of information on the system that was previously lost due to the presence of the environment.
Currently, an active field of research focuses on the investigation of the possible usefulness of memory effects for quantum technologies. A number of results, indeed, seem to support the idea that non-Markovian environments are most suitable for quantum communication and information processing purposes [12]. In particular, very recently it has been investigated in Ref. [13] how non-Markovianity affects the effectiveness of optimalcontrol strategies in the case of amplitude-damping-type channels, finding the existence of regimes where non-Markovianity can be either beneficial or detrimental. The results presented in this paper sit in the context above. We investigate the connection between the non-Markovian character of the open system dynamics and the efficiency of simple DD schemes. With a shift in perspective, we also study if and how the DD pulses change the non-Markovian character of the dynamics, e.g. whether they induce information back flow.
The structure of the paper is the following. In Sec. II we introduce the system of interest, namely the pure dephasing model including its exact solution in presence of periodic dynamical decoupling. In Sec. III, we discuss how the DD pulses affect information flow and hence modify the Markovian/non-Markovian character of the dynamics. In Sec. IV, we investigate whether non-Markovian or Markovian environment are best suited to DD, i.e., lead to optimal performance. In Sec. V, we discuss in the spirit of reservoir engineering, the optimum system parameters for achieving maximum stationary co-arXiv:1502.02528v1 [quant-ph] 9 Feb 2015 herences. Finally in Sec. VI we summarize our findings and draw the conclusions.

II. THE SYSTEM
Let us consider the following microscopic Hamiltonian describing the local interaction of a qubit [14] (i.e., a twolevel system) with a bosonic reservoir, in units ofh [15], with ω 0 the qubit frequency, σ z the usual z-component of the qubit pseudospin, ω k the frequency of the kth reservoir mode, a k (a † k ) the corresponding annihilation (creation) operator and g k the coupling constant associated with the qubit-kth mode interaction. This model can be solved exactly [16]- [18]. In the interaction picture, the master equation for the qubit density matrix ρ is given byρ the solution of which yields decay of the coherences (pure dephasing) as follows where γ 0 (t) = dΓ 0 (t)/dt (4) and Here, I(ω) = j δ(ω − ω j )|g j | 2 is the spectral density function characterising the interaction of the qubit with the oscillator bath (this is assumed to be at zero temperature). We consider the widely studied class of spectral densities of the form: with s the Ohmicity parameter and ω c a cutoff frequency. Ohmic spectra correspond to s = 1, super-Ohmic to s > 1 and sub-Ohmic to s < 1.
Let us now address the qubit behaviour in the presence of an arbitrary sequence of instantaneous bang-bang pulses, each of which being modelled as an instantaneous π-rotation around the Z-axis in the Bloch sphere. In such a case, the decoherence process of the qubit can still be exactly described by replacing Γ 0 (t) in Eq. (3) with a modified decoherence function Γ(t) [7]. An exact representation of the controlled decoherence function in terms of its free (uncontrolled) counterpart has been obtained in Ref. [7]. Consider an arbitrary storage time, t, during which a total number of N pulses are applied at instants As shown by Uhrig [19], the controlled coherence function Γ(t) can be worked out as, where, for 1 ≤ n ≤ N , In the next section, we will use Γ(t) as given above to investigate the dynamics in terms of information flow and quantum non-Markovianity.

III. PULSE-INDUCED INFORMATION FLOW REVERSAL
To characterize the open dynamics under study from the viewpoint of information flow, we make use of a well-known measure of non-Markovianity introduced by Breuer et al [11]. This is based on the time evolution of the trace distance between a pair of initial states of the open system, this being a measure of their relative distinguishability [14]. In a Markovian process the distinguishability between any two quantum states decreases monotonic ously in time, indicating a loss of information into the reservoir. In a non-Markovian process, in contrast, it can grow for some time intervals, indicating information back flow into the system. For the system here considered the non-Markovianity measure of Ref. [11] has a simple analytical expression [20]: where γ(t) [cf. Eq. (4)] is the modified decoherence rate and the integral, as suggested by our notation, is extended over the time intervals such that γ(t) < 0. Hence, one can immediately associate information backflow with negative values of γ(t). The non-Markovianity defined in this way for the free system has been studied previously (see Ref. [21]- [24]). In particular, it was shown analytically that, for Γ(t) = Γ 0 (t) and in the case of spectral density (6), the measure takes non-zero values if and only if s > 2 [9]. The controlled decoherence function Γ n (t) [cf. Eq. (8)] may be rewritten as: Hence, it is straightforward to define a relation connecting γ n (t n ), namely dΓ n /dt at the moment the system is pulsed t n , and the analogous value at the previous instant [7]: where for 1 ≤ n ≤ N , Interestingly, Eq.(11) clearly shows that information flow is reversed whenever a pulse interrupting the freesystem dynamics is applied. Hence, in terms of the system dynamics, a Markovian evolution is always changed into a non-Markovian evolution.
Let us now consider the time evolution of coherences of the purely dephasing system, undergoing periodic DD as shown in Fig. 1. For the sake of simplicity, we focus here on equally-spaced DD pulses applied at times t n = n∆t, with n = 1, 2, 3, .... We indicate witht the first time instant at which γ 0 (t) = 0, i.e., after which information flow is reversed (in the unperturbed dynamics). This time always exists for s > 2 [9]. For Markovian environments the shorter is the interval between the DD pulses, the higher is the efficiency of the DD scheme. For non-Markovian ones the same holds, provided that ∆t <t. We therefore focus our attention on this short-pulsing regime and ask the following question: is a Markovian or rather a non-Markovian unperturbed environment the most suited to achieve via DD the highest coherence preservation?
It is easy to see that, for any time t <t, the unperturbed coherences are always higher for s ≤ 2 (Markovian case) than for s > 2 (non-Markovian case). Since the effect of the pulses is always to reverse information flow and therefore preserve better coherences, we conclude that, in the short-pulsing regime, Markovian Ohmic environments lead to optimal DD compared to non-Markovian ones. Both in the Markovian and in the non-Markovian case, however, DD inhibits loss of coherence compared to the unpulsed free evolution. This is illustrated in Fig. 1 i.
We note for completeness that in the long-pulsing regime, defined by ∆t >t, the efficiency of the DD scheme here considered is drastically reduced and greatly depends on the details of the dynamics, hence no general conclusion can be drawn. It is worth noticing, however, that in this case reversing information flow can have disastrous consequences for non-Markovian environments. If the first pulse occurs during a time of re-coherence (information flowback), it will indeed induce a faster decay. This effect can be seen in Fig. 1 ii. This shows that DD does not work effectively for non-Markovian environments for large pulse spacing such that ∆t <t, and in fact leads to an even faster deterioration of coherences when compared to the unpulsed free dynamics.
To elucidate the relationship between the non-Markovian character of the free dynamics and the efficiency of dynamical decoupling techniques, we perfom a numerical investigation based on their respective measures. In order to link the sensitivity of DD performance to the form of the spectrum, we define the efficiency D of the protocol, bounded between zero (ineffective DD) and unity (ideal DD), as with t f the instant of the last applied pulse. In Fig. 2, we compare the DD efficiency measure D, as defined by Eq. (13) with Γ(t) given by Eq. (7), and the non-Markovianity N , as defined by Eq. (9) with free decoherence Γ 0 (t) given by Eq. (5), as functions of the Ohmicity parameter s. We focus here on the short pulse regime where the efficiency of the DD scheme is the highest. The plot clearly shows a sharp decrease in D with the onset of non-Markovianity for s > 2. This quantity, however, is only sensitive to the Markovian to non-Markovian crossover (s = 2) and not to the value of N for s > 2, as it keeps decreasing monotonically while N has a clear peak around s 3.7. For increasingly longer times t f , the efficiency becomes increasingly sensitive to the onset of non-Markovian dynamics, indeed, for t f → ∞, we conjecture D → 0 for s > 2.
Summarizing, the maximum efficiency of PDD is obtained for pulse spacings ∆t <t with Markovian environments. As the formalism used to describe the dynamics holds for any arbitrary Gaussian phase randomization process our conclusions hold in general for these types of models [7].

IV. NON-MARKOVIANITY ENGINEERING BY DYNAMICAL DECOUPLING
Markovian open quantum systems have been extensively studied and are very well characterized. For Markovian dynamics fulfilling the semigroup property [15], the Lindblad-Gorini-Kossakowski-Sudarshan theorem identifies the general form of master equation leading to a physical evolution of the system. The Monte Carlo wave function approach provides both a powerful numerical technique to study the dynamics of Markovian systems and a deep interpretation in terms of quantum jumps for individual quantum systems, like ions or cavity modes. Quantum state diffusion methods allow to unravel the dynamics in terms of homodyne or heterodyne measurements on the environment.
For non-Markovian open quantum systems many fundamental questions are still open. The generalisation of the Lindblad-Gorini-Kossakowski-Sudarshan theorem to even simple non-Markovian systems is still an open problem. The existence of a measurement scheme interpretation guaranteeing a physical meaning to individual trajectories is still under investigation. The extension of the Monte Carlo wave function approach is only known for certain classes of time-local master equations [25]. The first experimental studies aimed at characterizing non-Markovian dynamics have only recently been conducted [26]. This witnesses the interest in developing techniques for engineering non-Markovian dynamics to be used as testbeds for experimental and theoretical investigations.
The results of Sec. III show that, in addition to its traditional employment as a method to hamper decoherence, DD can be exploited as a simple tool for engineering non-Markovian dynamics. A Markovian open system will, indeed, always become non-Markovian when subject to PDD. More in general, PDD will change the non-Markovian character of the open system, whether its free dynamics was Markovian or not. Yet, the details of the pulse-induced non-Markovianity will depend on both the pulsing parameters (e.g., the pulse spacing) and the en-vironmental parameters (e.g., the Ohmicity parameter).
In this section, we investigate the non-Markovianity induced by PDD by comparing the non-Markovianity measure N in absence and presence of pulses, in both the short-pulse and the long-pulse regimes introduced in Sec. III. As we noticed there, in the short-pulse regime, for any value of the Ohmicity parameter s, the effect of the pulses is to create non-Markovinity by inducing information back flow, when it was initially absent, or in any case to increase the non-Markovian character. This can be seen in Figure 3 i. In the long-pulsing regime the situation is more variegated as pulses can also, under certain conditions, decrease the non-Markovian character of the dynamics, as shown in the example of Fig. 3 ii. We note in passing that in order to explicitly connect the way in which information flows with reservoir engineering, one can define analytically the timet in terms of parameters of the system.

V. LONG-TIME DYNAMICS
While in the previous sections we have shed light on the connection between the spectral density shape and the dynamical decoupling effectiveness for short times, we now turn our focus to the asymptotic behavior of the pulsed system. With reservoir engineering in mind, we investigate the connection between the stationary coherences and the form of the spectral density function (specifically, the value of the Ohmicity parameter s). More precisely, we study which value of s yields maximum long-time stationary coherences, for given pulses time spacing ∆t and number of pulses n.
We begin by noticing that in absence of pulses the phenomenon of coherence trapping occurs for s > 1, while for s ≤ 1 coherences are asymptotically lost as t → ∞. Fig. 4 shows how the maximum stationary coherence in the DD scheme depends on the pulsing interval as a function of s and for different numbers of pulses n. From Ref. [10] we know that, for the unperturbed system, the optimal Ohmicity parameter, i.e., the value of s leading to maximum long-time coherences, lies in the non-Markovian parameter range (s 2.46). In presence of pulses, this continues to be true, independently of the number of pulses n, only in the short-pulsing regime, as one can see by setting ω c ∆t = 0.3 (first column of dots in Fig. 4). We note that this is in contrast to what happens for short times. We have indeed noted in Sec. III that, in the short time dynamics, Markovian reservoirs lead to better coherence preservation when the interval between pulses is short. Hence the choice of optimal parameter is dependent on whether we are interested in short or asymptotically long time scales. As shown in Fig. 4, when the interval between pulses increases, Markovian reservoirs become better suited to long-time coherences preservation for most values of n, with the only exception of the somewhat special case n = 1 (red dots).

FIG. 4. (Color online)
Ohmic parameter s required to achieve maximum stationary coherence as a function of the (rescaled) pulse interval spacing ωc∆t for n = 1 (red dots), n = 2 (orange), n = 10, (green), n = 20 (blue), n = 60 (purple) and n = 80 (grey). The free Markovian (non-Markovian regime) corresponds to the shaded (unshaded) region, while the red line shows the value of the Ohmic parameter for the free stationary coherence.

VI. CONCLUSIONS
To conclude, our results provide indications on how one should engineer an environment which is optimal for dynamical decoupling techniques. We have explored the connection between information flow and dynamical decoupling to shed light on the phenomena responsible for revivals in the coherence. In more detail, we have shown that the coherence of the system is either revived by the free underlying non-Markovian dynamics or by the decoupling but never both of them, which is due to the pulse-induced reversal of the information flow direction. Having efficient error correction in mind, we have paid special attention to the short-pulses regime, i.e., when ω c ∆t 1. In this case, we have found that a Markovian environment is necessary to optimize the DD performance. However, the highest preserved coherence in the decoupling interval (short time dynamics) is not necessarily the highest stationary coherence (long time limit), i.e., the optimal Ohmicity parameter is not the same for the two regimes. Our work provides the first exploration of the interplay between information flow (non-Markovianity) and the efficiency of DD schemes. With a shift in perspective, it also indicates how dynamical decoupling techniques can be harnessed to engineer quantum non-Markovianity and control it.