Persistent quantum walks: dynamic phases and diverging timescales
aa r X i v : . [ qu a n t - ph ] S e p Persistent quantum walks: dynamic phases and diverging timescales
Suchetana Mukhopadhyay and Parongama Sen Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India.
A discrete time quantum walk is considered in which the step lengths are chosen to be either 1or 2 with the additional feature that the walker is persistent with a probability p . This implies thatwith probability p , the walker repeats the step length taken in the previous step and is otherwiseantipersistent. We estimate the probability P ( x, t ) that the walker is at x at time t and the firsttwo moments. Asymptotically, h x i = t ν for all p . For the extreme limits p = 0 and 1, the walk isknown to show ballistic behaviour, i.e., ν = 2. As p is varied from zero to 1, the system is found infour different phases characterised by the value of ν : ν = 2 at p = 0, 1 ≤ ν ≤ / < p < p c , ν = 3 / p c < p < ν = 2 again at p = 1. p c is found to be very close to 1 / p = 0 ,
1, the scaling behaviour shows a crossover in time. Associated with this crossover,two diverging timescales varying as 1 /p and 1 / (1 − p ) close to p = 0 and p = 1 respectively aredetected. Using a different scheme in which the antipersistence behaviour is suppressed, one gets ν = 3 / < p <
1. Further, a measure of the entropy of entanglement isstudied for both the schemes.
I. INTRODUCTION
Discrete time quantum walks (DTQW’s), first intro-duced by Aharonov et al .[1], are random walks where acoin degree of freedom is introduced which determinesthe translation of the walker. Quantum interference insuch walks leads to the position x of the walker scalingas h x i ∝ t with time t , indicating a quadratically fasterspread than the classical random walk.The introduction of randomness or disorder in quan-tum walks has been demonstrated to modify its scalingbehavior significantly. Disorder can be incorporated invarious ways, and several studies in recent years have fo-cused on the modifications they impart and how theymay turn out to be useful [2]. Disorder introducedthrough interaction with the external environment orthe presence of broken links in position space tends toslow down the quantum walk and leads to localization[3, 4]. Such localization effects were first studied by An-derson [5] in the context of electron localization in a dis-ordered lattice. Anderson localization type behavior hasbeen observed experimentally by introducing static (po-sitional) disorder in a quantum walk on a homogeneouslattice [6]. Static disorder can be contrasted with dy-namic disorder or decoherence [4] that transforms thequantum walk to the classical equivalent (diffusive scal-ing; h x i − h x i ∝ t ).Dynamic disorder is usually introduced through theoperations controlling the evolution of the quantum walk,such as by using decoherent coins [7, 8]. It is also possibleto incorporate dynamic disorder by relaxing the standardassumption of a constant displacement at each time stepand allowing longer steps to be chosen randomly, as in[9, 10] where the scaling h x i ∝ t was found.In the present work we introduce the concept of persis-tence in the quantum walk. In a classical random walk,persistence implies that the walker continues in the direc-tion taken in the previous step, making it non-Markovian.In the quantum walk, in order to introduce the idea ofpersistence, we allow the walker to take different step lengths at each step and remember the step length cho-sen in the preceding step with a certain probability. Thisprovides a simple way to study the effect of short termmemory in the long-ranged walk. The probability dis-tribution of the position of the walker and its first andsecond moments are evaluated and the results comparedwith the classical walk and the quantum walk withoutdisorder. In addition, we evaluate the entropy of entan-glement.Previously, a few studies have been made where mem-ory has been incorporated in different ways in a quantumwalk [11–15]. Both short term and long term memoryhave been considered but in a very different manner. Inparticular, in the case of the so-called ‘elephant quantumwalk’ [14] where the walker has infinite memory as well astime dependent step lengths, the variance was found toscale as t . To the best of our knowledge, the persistencein quantum walks, the way it has been incorporated inthe present work, has not been considered before.In the next section we introduce the quantum walkand the exact way the concept of persistence has beenused. In Section III, the results have been presented. Sec-tion IV includes a summary of the results and a detaileddiscussion on the implications and the insight developedthrough the study. II. THE PERSISTENT QUANTUM WALK
In the simple DTQW in one dimension, the walkercan occupy discrete, equispaced sites x on the real lineand takes a step at unit time intervals. In addition tothe position, the walker is assigned a second degree offreedom, by means of a coin state (either left ( | L i ) orright ( | R i ). The state of the walker is described by thefollowing two-component vector expressing probabilityamplitudes for the coin states: | ψ ( x, t ) i = h x | ψ ( t ) i = (cid:20) a ( x, t ) b ( x, t ) (cid:21) . (1)The occupation probability of the site x at time step t isgiven by P ( x, t ) = |h x | ψ ( t ) i| = | a ( x, t ) | + | b ( x, t ) | withthe total probability is equal to 1 at each time step. Astep in the quantum walk consists of a rotation in thecoin space followed by a translation. A standard choicefor this rotation operator is the Hadamard coin H , givenby H = 1 √ (cid:20) − (cid:21) . (2)Instead of defining the step length to be a constant l ,we allow it to be chosen from a binary distribution; l ( t ) = { , } at any given time t . The conditional translationoperator at a time t is then written as T ( t ) = | R i h R | ⊗ X x | x + l ( t ) i h x + l ( t ) | + | L i h L | ⊗ X x | x − l ( t ) i h x − l ( t ) | (3)Allowing for a non-unique step length in this way enablesone to study the phenomenon of persistence by consid-ering the tendency to adopt the step length used in theprevious time step. This choice is made in two ways,outlined under two different schemes, I and II. In eachscheme, at t = 0, the step length l (0) = 1 or l (0) = 2is chosen with equal probability. In Scheme I, at anylater time t = 0, the walker either chooses the same steplength as in the previous time step (persistent) and oth-erwise necessarily chooses the other step length (anti-persistent). In Scheme II, the walker is persistent witha probability p , but this time with probability (1 − p ),either of the step lengths l = 1 and l = 2 are chosen,with probability q and (1 − q ) respectively.For both schemes the walker is initialised with a ( x,
0) = b ( x,
0) = √ δ x, which gives an asymmetricprobability distribution profile in the absence of disor-der. The walk is evolved for 20000 time steps for allparameter values. We investigate how the occupationprobability, moments, and entanglement depend on theparameter(s) used in the two schemes. All results areaveraged over 4000 configurations. III. RESULTSA. Scheme I
In this subsection we present the results for the firstscheme considered. The walker here chooses the steplength taken in the previous step with probability p andwith probability (1 − p ) it chooses strictly the otherlength. The latter case thus corresponds to an antiper-stent choice.
1. Probability distribution
When p = 0, steps of length l = 1 and 2 are takenalternately, with a possible sequence of steps given by1 , , , , ... etc. The walk clearly has periodicity 2, andthere is no randomness in the choice of steps. This partic-ular walk has already been studied in [9] and was found tohave the same scaling behavior as the ordinary quantumwalk. The probability distribution resembles an over-lap of distributions obtained for the ordinary walks with l = 1 and 2 [9]. -5 -4 -3 -2 -1 -200-150-100 -50 0 50 100 150 200p=0.1 P ( x , t ) t (cid:1) x/t (cid:0) -3 -2 -1 -1.5 -1 -0.5 0 0.5 1 1.5p=0.1 P ( x , t ) t γ x/t γ -5 -4 -3 -2 -1 -200-150-100 -50 0 50 100 150 200p=0.5 P ( x , t ) t γ x/t γ -3 -2 -1 -1.5 -1 -0.5 0 0.5 1 1.5p=0.5 P ( x , t ) t γ x/t γ -5 -4 -3 -2 -1 -200-150-100 -50 0 50 100 150 200 p=0.95 P ( x , t ) t γ x/t γ -3 -2 -1 -1.5 -1 -0.5 0 0.5 1 1.5 p=0.95 P ( x , t ) t γ x/t γ FIG. 1. Scheme I : Data collapse of rescaled P ( x, t ) using γ = 0 . γ = 1 . γ = 1 collapseis less sharp. On the other hand, when p = 1, the walk becomesdeterministic with a unique step length and thus identicalto the usual quantum walk. The result for the point p =0 . p is increased even slightly from zero, the distri-bution P ( x, t ) exhibits a peak centered at the origin inaddition to two ballistic peaks. Such a central peak isnot present in the ordinary quantum walk, neither in thebinary walk without randomness. However, in presenceof disorder and decoherence, such peaks indeed appear,when localisation of the quantum walker takes place.The ballistic peaks are signatures of the quantum na-ture of the walk, and are in general asymmetric in height,reflecting the asymmetry of the pure quantum walk. As p is increased, the ballistic peaks are seen to increase inheight, while the central peak goes down. This can be in-
100 1000 10000 <(cid:2)(cid:3) tp=0p=0.005 p(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) p=0.95tt
FIG. 2. Scheme I: The first moments for four p values showsignificant change in behaviour as p is varied. terpreted as an increase in delocalisation with increasingprobability of the walker to be persistent, as the walkerapproaches the standard case of a constant step length.Even for p very close to one, however, the central peakdoes not disappear. Exactly at p = 1, the familiar distri-bution of the quantum walk is recovered, as expected.Plotting t γ P ( x, t ) against the scaled variable x/t γ , weobserve data collapse for four different time steps, with γ = 0 . γ = 1 . P ( x, t ) exhibits two dis-tinct scaling behaviors: the centrally peaked part scalesas x ∝ √ t , similar to the classical walk, while the bal-listic peaks scale like the ordinary quantum walk, x ∝ t .These results are true for the entire region 0 < p <
2. Scaling of the moments
We next present the results for the first two moments, h x i and h x i of the probability distribution.In the limiting cases of p = 0 and p = 1, the walkreduces to an ordinary quantum walk such that h x i ∝ t and h x i ∝ t . Interestingly, when p deviates from zeroor one by even the smallest amount, we note that theasymptotic variations of the moments are significantlychanged. The first moments are plotted in Fig. 2 fora few p values, showing that the asymptotic exponentdecreases to very small values for p = 0 + and increasesup to a value close to 1/2 at larger p values.We plot the second moments in Fig. 3 for a few valuesof p . Usually it is the variance which is used to char-acterise the walk. However, we note here that asymp-totically, h x i is larger than h x i by at least two ordersof magnitude. Hence it suffices to consider how h x i be-haves instead of the variance h x i−h x i . In the following,we study the behaviour of the second moment in detail.We note two things from Fig. 3; first, for p values closeto zero 0 and 1, there is a distinct change in the behaviourof the moments with time; initially it has a fast growth < x tp=0.005p=0.35p=0.95tt FIG. 3. Scheme I: The second moment for three p val-ues. The continuous lines are best fit curves obtained us-ing Eq. 5 with fitting parameters α ′ = 1 . , . , .
81 and β ′ = 0 . , . , .
066 for p = 0 . , .
35 and 0 .
95 respec-tively. < x > /t pt p=0.00025p=0.0005p=0.00075p=0.001 10 < x > t x p=0 FIG. 4. Scheme I: Data collapse of the second moment ofthe distribution P ( x, t ) obtained on plotting h x i / t againstthe scaling variable pt . The solid line is a best fit line drawnusing Eq. 4. Inset shows the unscaled data h x i against time t . The data for p = 0 and a curve with quadratic variationare shown for comparison. but becomes slower later. Secondly, the asymptotic vari-ation is significantly dependent on p ; for a value of p veryclose to zero, the exponent is close to 1. This is a drasticchange from the value 2 when p is exactly zero.We probe the p → + region in more detail as we havethe most significant change in the asymptotic exponentvalue here. Here, h x i plotted in the inset of Fig. 4,clearly shows a variation compatible with t for a longtime before deviating to a slower variation. The devia-tion occurs at larger values of time as p approaches zero.Plotting h x i / t against the scaling variable pt for sev-eral values of p very close to zero, we obtain a very goodcollapse (Fig. 4) from which we claim h x i ∝ t f ( pt ). Itis clear from Fig. 4 that f ( z ) fairly a constant for z < z ≥
1, one can fit f ( z ) to the form f ( z ) = 1 / ( α + βz µ ) (4)to a great degree of accuracy with α ≈ . , β ≈ . µ ≈
1. From this it can be concluded that a crossoveroccurs at pt ≈ p . On the other hand the asymptoticvariation is h x i ∝ t . Thus the crossover time marks thetransition to the asymptotic behaviour.As p is made larger, the crossover occurs at smallertimes and the exponent is extracted from fitting the sec-ond moment directly to the empirical form valid for latertimes: h x i = t / ( α ′ + β ′ t − ν ) , (5)such that asymptotically, h x i ∝ t ν . The best fit curvesusing the above form plotted for the data shown in Fig.3 show excellent agreement. In Fig. 5, the asymptoticexponents ν is plotted as a function of p . One notes thatthe exponent continuously varies from 1 to 3 / < p < p c where p c is approximately 0 . p = 1, where alsothe exponent shows a jump from the value 2 to ∼ / − p ) t such that the associ-ated timescale diverges as − p .We note therefore that the walk shows a superdiffusivebehaviour but with a nonuniversal exponent for small p where the antipersistent effect is strong. For largervalues of p , the walker behaves as that with random steplengths and persistence is apparently merely the tool thatprovides the stochasticity. Since it was found in [9] thatthe slightest randomness alters the exponent to 3/2, it isnot surprising to see that for large p = 1, one gets thesame exponent. (cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26) (cid:27) (cid:28)(cid:29)(cid:30) (cid:31)!" p Scheme IScheme II; q=0.25Scheme II; q=0.5
FIG. 5. Variation of the effective scaling exponent ν with p for Scheme I and Scheme II.
3. Entanglement entropy
In any quantum walk, the evolution operator generatesentanglement between the position and coin degrees offreedom. This entanglement can be quantified using the +,-./01345679 : ;=? @AB CDE FGH IJKLMNOS P ( t ) TQRU VWXY Z[\]^_‘abcdefghij klmnoqrstuvwx y z{| }~(cid:127) (cid:128)(cid:129)(cid:130) (cid:131)(cid:132)(cid:133) (cid:134)(cid:135)(cid:136)(cid:137)(cid:138)(cid:139)(cid:140)(cid:141)(cid:142) ( t ) (cid:143)(cid:144)(cid:145)(cid:146) (cid:147)(cid:148)(cid:149)(cid:150) (cid:151)(cid:152)(cid:153)(cid:154)(cid:155)(cid:156)(cid:157) (cid:158)(cid:159)(cid:160)¡¢£⁄¥ƒ§¤' “«‹›fifl(cid:176)–†‡· (cid:181)¶•‚„”»…‰(cid:190)¿ (cid:192)`´ˆ˜ FIG. 6. Entanglement entropy plotted against time forScheme I (a) and Scheme II (b). von Neumann entropy S E ( t ), also known as the entropyof entanglement. This can be evaluated from the reduceddensity operator which is represented by the followingmatrix [16] ρ c ( t ) = (cid:20) A ( t ) B ( t ) B ( t ) C ( t ) (cid:21) (6)where we have A ( t ) ≡ P x | a ( x, t ) | ; B ( t ) ≡ P x | a ( x, t ) || b ( x, t ) | ; C ( t ) ≡ P x | b ( x, t ) | . The entropyof entanglement S E ( t ) is then calculated as S E ( t ) = − T r ( ρ c log ρ c )= − ( v log v + v log v ) . (7)where v and v are the real, positive eigenvalues ofthe matrix ρ c ( t ). We numerically evaluate S E ( t ) forthe quantum walk with the localised initial condition a (0 ,
0) = √ , b (0 ,
0) = √ , taking 0 log p = 1, S E ( t ) is found toasymptotically converge to ≈ . ... for our chosen ini-tial condition, as has been previously reported [16–18].For p = 0, we obtain S E ( t ) ≈ . ... . As the walk devi-ates from either of the two extremes, the value of S E ( t )increases drastically, rapidly converging to a large valuevery close to unity for any p . Although the parameter p does not significantly influence the limiting value (atleast not up to three decimal places), the rate of con-vergence is faster for smaller p values. Fig. 6 shows thebehavior of the entanglement for a few values of p . B. Scheme II
This scheme represents a variation of the first wherethe walker is either persistent with probability p or, withprobability (1 − p ) can choose step length l = 1 or 2with probability q and (1 − q ) respectively. Obviously,for p = 1 it is always persistent and one recovers anordinary quantum walk. For p = 0, it takes step lengths1 and 2 randomly unless q = 0 or 1. In fact, if q = 0or 1, the walk becomes of unique step length eventually,(independent of p ); if q = 1, that length is 1 and 2 for q = 0.Effectively, the total persistence probability p ′ of thewalker in this scheme is either p + q (1 − p ) or p + (1 − q )(1 − p ) at a given time step and it is antipersistentwith probability 1 − p ′ . When q = 0 .
5, the walker is thuspersistent with a probability p ′ = p/ / q , and the results should correspond to those obtainedfor a persistence probability p ′ in Scheme I. Since p ′ > p c ,one can expect that the results for Scheme II will beidentical to Scheme I with the second moment scaling as t / asymptotically for all p ≥
0. Even when q = 0 .
5, thewalk is persistent with probability p/ / p ′ ), when ensemble average is taken if thefluctuation is negligible. In the following we discuss theresults which confirms the above picture. -5 -4 -3 -2 -1 -200-150-100-50 0 50 100 150 200 ¯˘˙¨(cid:201)˚ ¸(cid:204)˝˛ˇ— P ( x , t ) t γ x/t γ -3 -2 -1 -1.5 -1 -0.5 0 0.5 1 1.5 (cid:209)(cid:210)(cid:211)(cid:212)(cid:213)(cid:214) (cid:215)(cid:216)(cid:217)(cid:218)(cid:219)(cid:220) P ( x , t ) t γ x/t γ -5 -4 -3 -2 -1 -200-150-100 -50 0 50 100 150 200 (cid:221)(cid:222)(cid:223)(cid:224)Æ(cid:226) ª(cid:228)(cid:229)(cid:230)(cid:231)Ł P ( x , t ) t γ x/t γ -3 -2 -1 -1.5 -1 -0.5 0 0.5 1 1.5 ØŒº(cid:236)(cid:237)(cid:238) (cid:239)(cid:240)æ(cid:242)(cid:243)(cid:244) P ( x , t ) t γ x/t γ -5 -4 -3 -2 -1 -200-150-100 -50 0 50 100 150 200 ı(cid:246)(cid:247)łøœ ß(cid:252)(cid:253)(cid:254)(cid:255) P ( x , t ) t γ x/t γ -3 -2 -1 -1.5 -1 -0.5 0 0.5 1 1.5 p(cid:0)(cid:1)(cid:2)(cid:3)(cid:4) (cid:5)(cid:6)(cid:7)(cid:8)(cid:9) P ( x , t ) t γ x/t γ FIG. 7. Scheme II: Data collapse of rescaled P ( x, t ) using γ = 0 . γ = 1 . γ = 1collapse is less sharp. Once again we obtain the distribution P ( x, t ) whichshows a peak centered at the origin and two ballisticpeaks. As seen for Scheme I, data collapse is observedwith γ = 0 . γ = 1 . x ∝ √ t and x ∝ t respectively.Next we show the variation of the second moment inFig. 8 against time which shows the unique exponent 3 / < p <
1. Here we have plottedthe results for both small and large values of p, q showingno significant difference. The exponent ν as a functionof p for q = 0 . q plotted in Fig.5 shows that it is 3/2 for the entire region 0 < p < p c .Lastly, we plot S E ( t ) against time t for chosen values of p and q (Fig. 6) which does not show any distinguishing < x t (cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31)!" FIG. 8. Scheme II: The second moment for a combination ofvalues of p and q . The continuous lines are best fit curvesobtained using Eq. 5. feature from Scheme I. IV. SUMMARY AND DISCUSSIONS
In the present work we have reported the results ona non-Markovian quantum walk where the step lengthsare binary at each time step. It is non-Markovian in thesense that the walker remembers the step length takenin the previous step and tends to repeat it with proba-bility p . Thus the choice of the step length is entirelydetermined by the value of p . We numerically evalu-ate the time evolution of the walk and calculate the dis-tribution P ( x, t ) and its moments. Scheme I, which isthe case when it is strictly antipersistent with proba-bility (1 − p ), leads to some non-intuitive results when p is small. Precisely, we find how the asymptotic be-haviour of the second moment changes as p is varied.One can locate four different phases through which ν changes. The first is the point p = 0, where the walkis periodic and correlated over infinite time range and ν = 2. At p = 0, there is however, a finite disconti-nuity in ν as ν = 1 for p = 0 + . For 0 < p < p c , ν shows a continuous increase with p . Here the walkdeviates from its periodic nature and the step lengthsare e.g., 1 , , , , ..... , , , , , , ......, , , , , ..... etc.,such that it has two “opposite” patterns repeating alter-nately. However, the long time correlation is weakened.As p increases, the step lengths tend to repeat, however,none of the strings, either e.g., 1 , , ... , , , , , p is not equal to 1or zero. As a result, one gets approximately a randomsequence of step lengths. In fact at p = 0 . p is exactly equal to 1 so that we get ν = 3 / p c < p < ν = 2 again at p = 1.The result for the region 0 < p < p c is perhaps themost interesting where we find a p dependent value of ν < / p regarding the step lengths 1 and 2as up/down states of a Ising spin. The two sequences1 , , , .... and 2 , , , .... are like antiferromagnetic pat-ters and are equivalent to simply spin flipped versions ofone another in this picture. Hence close to p = 0 we havetwo alternating antiferromagnetic patterns and close to p = 1 we will have two alternating ferromagnetic patternsseparated by domain boundaries. Interestingly for p → p → p = 1, theconfinement comes only from the fact that the repetitionof the two ferromagnetic patterns is in no way periodicin nature. However, why the antiferromagnetic and fer-romagnetic sequences lead to different scaling behaviourand why p c is close to 1/3 remain issues to be resolved.One other result is that in general the first moment showsa scaling h x i ∝ t ν − which is not quite obvious.The second important result we obtain is the crossoverphenomena near p = 0 and 1. Simple power law scalingsfor the moments are not possible here as clearly the be-haviour changes in time. Of course, one can continue thenumerical evolution for even larger number of time steps and extract the asymptotic variation. In practice, it isbeyond the computational capacity to do so. However,identification of the scaling variable and consequently ob-taining a form of the scaling function could help in cal-culating the asymptotic exponent. We could detect thepresence of timescales which diverge at the extreme limitsand in the process reveal that the crossover phenomenatakes place here with a diverging time scale. This di-vergence signifies that although the exponent ν changesdiscontinuously as p → p →
1, the change can beobserved only after long time scales. Away from the ex-treme limits, the crossover effect becomes less conspicu-ous.The results for the persistent quantum walker showsthat it is clearly different from the case of random choiceof step lengths as long as antipersistence is strong, result-ing in a nonuniversal value of the exponent. This is allthe more evident from the results of Scheme II in whichthe effective persistence probability corresponds to thatin Scheme I with p > / ν = 3 /
2. Although the scaling expo-nent ν is p dependent for p < p c , the P ( x, t ) data showcollapse with the same type of rescaling as in the randomcase. Also, the results for the entanglement entropy arequalitatively similar to that of the latter case.Acknowledgement:SM is grateful for the opportunity to work in the De-partment of Physics, University of Calcutta. PS is grate-ful to SERB scheme number: EMR/2016/005429. [1] Y. Aharonov, L. Davidovich, and N. Zagury,Phys. Rev. A , 1687 (1993).[2] V. Kendon, Mathematical Structures in Computer Sci-ence , 1169 (2007).[3] J. P. Keating, N. Linden, J. C. F. Matthews, and A. Win-ter, Phys. Rev. A (2006).[4] Y. Yin, D. E. Katsanos, and S. N. Evangelou,Phys. Rev. A , 022302 (2008).[5] P. W. Anderson, Phys. Rev. , 1492 (1958).[6] A. Schreiber, K. N. Cassemiro, V. Potoˇcek,A. G´abris, I. Jex, and C. Silberhorn,Phys. Rev. Lett. , 180403 (2011).[7] T. A. Brun, H. A. Carteret, and A. Ambainis,Phys. Rev. Lett. , 130602 (2003).[8] T. A. Brun, H. A. Carteret, and A. Ambainis, Phys.Rev. A , 032304 (2003).[9] P. Sen, Physica A , 266 (2019). [10] P. Sen, (2019), arXiv:1902.09129 [quant-ph].[11] M. McGettrick, Quantum Information and Computation , 0509 (2010).[12] P. P. Rohde, G. K. Brennen, and A. Gilchrist,Phys. Rev. A , 052302 (2013).[13] D. Li, M. Mc Gettrick, F. Gao, J. Xu, and Q.-Y. Wen,Phys. Rev. A , 042323 (2016).[14] G. Di Molfetta, D. O. Soares-Pinto, and S. M. D.Queir´os, Phys. Rev. A , 062112 (2018).[15] M. A. Pires, G. D. Molfetta, and S. M. D. Queirs, (2019),arXiv:1907.12696 [quant-ph].[16] G. Abal, R. Siri, A. Romanelli, and R. Donangelo, Phys.Rev. A , 042302 (2006).[17] I. Carneiro, M. Loo, X. Xu, M. Girerd, V. Kendon, andP. L. Knight, New Journal of Physics , 156 (2005).[18] G. Abal, R. Siri, A. Romanelli, and R. Donangelo,Phys. Rev. A73