High Order Coherent Control Sequences of Finite-Width Pulses
aa r X i v : . [ qu a n t - ph ] S e p High Order Coherent Control Sequences of Finite-Width Pulses
S. Pasini, P. Karbach, and G. S. Uhrig ∗ Lehrstuhl f¨ur Theoretische Physik I, Technische Universit¨at Dortmund,Otto-Hahn Straße 4, 44221 Dortmund, Germany (Dated: May 21, 2018)The performance of sequences of designed pulses of finite length τ is analyzed for a bath of spinsand it is compared with that of sequences of ideal, instantaneous pulses. The degree of the designof the pulse strongly affects the performance of the sequences. Non-equidistant, adapted sequencesof pulses, which equal instantaneous ones up to O ( τ ), outperform equidistant or concatenatedsequences. Moreover, they do so at low energy cost which grows only logarithmically with thenumber of pulses, in contrast to standard pulses with linear growth. PACS numbers: 03.67.Pp, 82.56.Jn, 03.67.Lx, 76.60.Lz
The rapid evolution of the field of quantum science andquantum information demands robust quantum controltechniques in the presence of environmental noise. To dy-namically generate systems essentially free from decoher-ence has now become a focus of the research of quantumcontrol. This suppression of decoherence is an importantrequisite in quantum information processing [1], for ex-ample for the realization of a quantum computer, in nu-clear magnetic resonance (NMR), for high accuracy mea-surements [2] or in magnetic resonance imaging (MRI)[3], to mention only a few.In this work we focus on quantum control by shortpulses of finite length. It is beyond our scope to discusscontinuous quantum control, see for instance Ref. 4. Itis on a discovery in NMR, the Hahn spin echo [5], thatthe pulsed-control methods are based. The original tech-nique makes use of an electromagnetic pulse in order torotate the spin and to refocus it along a desired direction.Dynamical decoupling (DD) [6, 7] iterates the single pulsein a sequence of pulses such that the coupling betweenthe spin and its environment is averaged to zero. Amongthe “open-loop” pulse-control techniques, the dynamicaldecoupling is one of the most promising protocols for pro-longing the coherence time of a spin (qubit) coupled toan environment. No detailed, quantitative knowledge ofthe decohering environment is required.The sequences come in a large variety. We distinguishequidistant and non-equidistant sequences. In the firstcategory we recall the iterated Carr-Purcell-Meiboom-Gill (CPMG) sequence [8, 9], where the pulses are reg-ularly separated (apart from the very first and the verylast one). To the second category belong for instancethe universal Uhrig DD (UDD) sequence [10–12], theLocally Optimized Dynamical Decoupling (LODD) [13],the Optimized Noise Filtration by Dynamical Decoupling(OFDD) [14] and the Bandwidth-Adapted DynamicalDecoupling (BADD) [15] for pure dephasing models andthe concatenated DD (CDD) [16] or UDD (CUDD) [17]or the quadratic UDD (QDD) [18] for models with de-phasing and relaxation.The design of the DD schemes relies originally on the assumption that the pulses are arbitrarily strong and in-stantaneous though the effects of pulses of finite lengthwere known to matter [19–23]. But the pulses used inlaboratories always have a bounded, finite amplitude sothat they have a finite duration τ . Even if sequenceslike CPMG and UDD have already been implemented inexperiments with very good results [3, 13, 24], the factthat pulses have a finite duration appears often as a nui-sance deteriorating the suppression of decoherence, seefor instance Refs. 20 and 22.It is of great practical relevance to which extent thelength of a pulse affects the performance of a sequencesuch as UDD or CPMG of given duration T . How shouldone choose the location, the duration (or the amplitude),and the shape [21, 23, 25] of the bounded pulse in order tominimize the errors due to its finite duration if it replacesthe ideal, instantaneous pulses in a certain sequence?Here we report for the first time numerical evidenceof how sequences of realistic pulses of finite width mustbe designed in order to achieve the same perturbativesuppression of dephasing as the corresponding ideal se-quence. We compare various known sequences [20, 26, 27]and numerically analyze their performance for a spin cou-pled to a bath of spins. To obtain an experimentally rel-evant comparison all pulses are designed in such a waythat the largest amplitude appearing in each sequence isthe same [28]. The Model.
We consider the pure dephasing Hamil-tonian H = 1 q ⊗ B + σ z ⊗ B z that determines the freeevolution of the system between two consecutive pulsesby U free ( t ) = exp {− itH } . The operators B and B z acton the bath only, while the identity 1 q and the Pauli ma-trix σ z act on the qubit represented by a spin 1 /
2. Forsimplicity we identify henceforth 1 q ⊗ B and B . Thebath consists of M spins with i ∈ { , . . . , M } H = ω b B + σ (0) z N s X i =1 λ i σ ( i ) z . (1)No drift term ∝ σ z of the qubit is included because wework in the rotating reference frame. Explicitly we an-alyze two cases, see also Fig. 1: (i) A spin chain with B = P Mi =1 ~σ ( i ) · ~σ ( i +1) , λ i ≡ λ and N s = 1. (ii) Acentral spin model [11, 29–33] characterized by a dipolarcoupling [2] B = P Mj
1) = λ (2 i − M −
1) and N s = M . The rapidityof the dynamics of the bath is given by ω b := αλ with α a dimensionless constant. central spin model:...012 3 40 2 31spin chain: 4 ... -10010 nd order π pulse -4-2024 SCORPSE t [ τ ] -20-1001020 v ( t ) [ τ - ] nd order 2 π pulse FIG. 1: Left: Spin bath models under study with a qubit(square) coupled (dashed lines) to M bath spins (dots) inter-acting among themselves (solid lines). Right: Upper panel:1 st order SCORPSE π pulse [34]; middle panel: 2 nd order π pulse; lower panel: 2 nd order 2 π pulse. Both 2 nd order pulsesare found by solving the conditions derived in Ref. 25. Am-plitudes and switching instants are available upon request. The control Hamiltonian is given by H c ( t ) = σ x v ( t ).We consider piecewise constant pulses shown in Fig. 1.During each pulse of total length τ ( i ) the qubit evolvesunder the simultaneous action of the system and ofthe control Hamiltonian U p = T exp {− i R t + τ ( i ) t ( H + H c ( t )) dt } where T stands for standard time ordering.The evolution operator of the total sequence from t = 0to t = T is denoted by b R . The Sequences.
Two types of sequences are studied,see also Fig. 2: (i) The durations τ ( i ) = τ ∗ of the pulsesis constant throughout the sequence and it is kept con-stant on variation of T . These sequences are denoted by τ j CPMG, τ j CDD and τ j UDD, because they reproducethe ideal CPMG, CDD and UDD sequences for τ → j stands for properties of the pulses as ex-plained below. (ii) The durations τ ( i ) are varied alongthe sequence, i.e., they depend on i . But they shall notdepend on T other than that the sum of all pulse du-rations cannot exceed T , i.e., T ≥ T p := P i τ ( i ) . Thecorresponding sequences are denoted by τ j RUDD.The sequences of type (i) are made of
N π pulses whosewidth is τ ∗ . The center of the i -th pulse is given by t CPMG i := T (2 i − / (2 N ) , (2a) t UDD i := T sin ( π i/ (2( N + 1))) , (2b)for the τ j CPMG and τ j UDD [10] sequence, respectively.We use the simplified version of CDD designed only for max Tt i UDD tt i − i + τ a max ∗ τ ∗ τ ∗ τ ∗ τ FIG. 2: (Color online) Upper panel: Sequences of type (i)( τ j CPMG, τ j CDD or τ j UDD) are sketched. Lower panel: Se-quences of type (ii) are shown ( τ j RUDD). Only the maximumamplitude and the pulse duration are shown, but no detailsof the pulse shapes. The π -pulses are depicted by filled (red)blocks, the initial and final 2 π -pulses by open (blue) blocks.The first π pulse of τ j RUDD and all pulses of type (i) se-quences have the same amplitude a max to ensure experimen-tally relevant comparability. The instants t i are given in Eqs.2 and 3; the start and end points t ± i in Eq. (4). pure dephasing. The CDD sequence of level k is definedby the recursionCDD k +1 ( T ) = CDD k ( T / ◦ Π π ◦ CDD k ( T / , (3a)CDD k +1 ( T ) = CDD k ( T / ◦ CDD k ( T / , (3b)where (3a) holds for k even and (3b) for k odd; ◦ standsfor concatenation and Π ϕ for the operator of a pulse ofangle ϕ . The zero-level CDD ( T ) is free evolution with-out pulses.The subscript j in τ j refers to the order of thepulses, i.e., its time evolution operator fulfills U p =exp {− iτ B } Π ϕ + O ( τ j +1 ). We restrict our study hereto explicit pulses with j = 0 , ,
2, see Fig. 1, which fulfillthe conditions derived in Ref. 25. A recursion for gen-eral j is given in Ref. 35. The 0 th order pulse is simplyrectangular; the other pulses used are depicted in Fig. 1.The sequences of type (ii) are similar to the τ j UDDsequences in that they are based on pulses of order j .The crucial difference is that pulses are not constant inlength. They are defined according to our previous work[27] by a start instant t − i and a stop instant t + i given by t ± i := T sin (cid:18) π i N + 1) ± θ p ( T )2 (cid:19) . (4)The above relation results naturally from the requirementthat the effective switching function of the sequence ex-pressed in θ ∈ [0 , π ] according to t = T sin ( θ/
2) is an-tiperiodic [27]. This antiperiodicity ensures that the to-tal sequence suppresses the decohering terms ∝ σ z in thetime evolution [12]. The duration of the pulses in time τ ( i ) = t + i − t − i yielding τ ( i ) = T sin ( π i/ ( N + 1)) sin( θ p ) (5)is determined by the parameter θ p ( T ). It acquires a de-pendence on T if we require τ ∗ := τ (1) to be constantupon varying T . Note that θ p = π/ (2( N + 1)) refers toback-to-back pulses without any free evolution betweenthem, see below.The antiperiodicity of the switching function is the ba-sis for the suppression of dephasing in high order [12, 36].In order to guarantee this antiperiodicity, it is requiredto insert an initial and a final pulse which represent theidentity U p = exp {− iτ B } + O ( τ j +1 ). For instance, itmay be a zero π or a 2 π pulse [27]. The initial pulse startsat t − = 0 and stops at t +0 = T sin ( θ p /
2) while the fi-nal one starts at t − N +1 = T sin [( π − θ p ) /
2] and stops at t + N +1 = T . These pulses are indicated by open boxes inFig. 2.In the sequel, we compare the various sequences alwayswith the same τ ∗ because the shortest accessible pulseduration of a π pulse, corresponding to the maximumamplitude, represents a crucial experimental constraint[15, 28]. Only the very short boundary 2 π pulses in the τ j RUDD are treated separately. But their importance isassessed by considering τ j RUDD with and without theboundary 2 π pulses. We stress that due to the variableduration of the pulses according to (4,5) in the RUDDsequence most of the pulses are much longer than τ ∗ . The Partial Frobenius ( ∆ pF ) Distance defines thedistance between the ideal evolution of the initial stateof the qubit due to the pulses and its evolution includ-ing the interaction with the bath and the application ofthe sequence [37]. For each axis of rotation γ = { x, y, z } we define a difference of density matrices of the qubit by ρ ( γ )q := tr B h ρ ( γ )id − ρ ( γ )qB i , where ρ ( γ )qB := b Rρ ( γ )0 b R † . Thepartial trace over the bath is denoted by tr B . Given afactorized initial state ρ ( γ )0 := | γ ih γ | ⊗ B the density ma-trix ρ ( γ )id := σ Nx ρ ( γ )0 σ Nx is the ideally evolved ρ subjectonly to ideal pulses without any bath interaction. Thedistance ∆ pF measures the difference between the realevolution and the ideal one reading∆ := 13 X γ = x,y,z tr q h ρ ( γ )q i . (6) Numerical Simulation.
We compute the performanceof sequences of pulses of finite duration for the systemsin (1) shown in Fig. 1. We choose the minimum duration τ ∗ < min i { τ ( i ) } and a minimum value of T such that T ≥ P i τ ( i ) , see captions for values. Sequences with N = 10 pulses are considered because this number allowsus to consider the CDD sequence as well; it correspondsto the concatenation level k = 4, cf. Eq. (3). The resultsare shown in Figs. 3 and 4(a) for the spin chain modeland in Figs. 4(b) and Fig. 5 for the central spin model.First, we consider the influence of the topology andthe size of the spin bath. In Fig. 3(c) data for the spinchain is shown for M = 3 (open symbols) and data for M = 8 (filled symbols) fits in perfectly. This indicates -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 ∆ p F τ CPMG τ CDD τ UDD τ RUDD (a) ∆ pF ∝ Τ spin chainM=3 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 ∆ p F τ CPMG τ CDD τ UDD τ RUDD (b) ∆ pF ∝ Τ spin chainM=3 T [1/ λ ] -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 ∆ p F τ CPMG τ CDD τ UDD τ RUDD CPMG CDD UDD (c) ∆ pF ∝ Τ spin chainM=3 & 8 FIG. 3: (Color online) Distance ∆ pF vs. the duration T ofsequences of pulses with zero or finite width for α = 10 forspin chains. All open symbols refer to M = 3 bath spins; thefilled symbols in panel (c) to M = 8 bath spins. The finite-width pulses have minimum width τ ∗ = 1 . · − /λ and T ≥ . /λ ; Panel (a): rectangular 0 th order pulses; Panel(b): SCORPSE 1 st order pulses [34]; Panel (c): 2 nd orderpulses shown in Fig. 1. To highlight power-law behavior thedashed lines are included: The UDD curve scales as T N +1) . that the size effect is very small in the regime of interest.The topology of the spin bath has a certain impact, butonly on the quantitative level, not on the qualitative oneas can be seen comparing Fig. 3(c) with Fig. 4(b). Theresults for the central spin model with M = 8 bath spinsare qualitatively identical to the ones for the spin chainexcept for a heuristic factor κ ≈ . T . The lattercan easily be understood in the sense of an effectivelystronger coupling between qubit and bath for the centralspin model than for the spin chain for the same value λ because there are more couplings λ i ∝ λ between qubitand bath spins.Second, we study the influence of the sequences onthe performance. Thus we consider long sequence dura-tions T . In this regime the pulse errors are unimportantand pulse shaping plays only a minor role. This fact isperfectly understandable because for given τ ∗ the limit T [1/ λ ] -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 ∆ p F τ CPMG τ CDD τ UDD τ RUDD CPMG CDD UDD (a) ∆ pF ∝ Τ spin chainM=3 T [1/ λ ] -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 ∆ p F τ CPMG τ CDD τ UDD τ RUDD CPMG CDD UDD ∆ pF ∝ Τ central spin M=8(b) FIG. 4: (Color online) Same as in Fig. 3(c). Panel (a): spinchain with M = 3, α = 100 and τ ∗ = 1 . · − /λ , T ≥ . /λ . Panel (b): central spin model with M = 8, α = 10and τ ∗ = 0 . /λ , T ≥ . /λ . The result is identical toFig. 3(c) except for a shift by the factor κ ≈ . T . T → ∞ implies that τ ∗ /T vanishes. In the formalismof filter functions [10, 13, 38–40] this can easily be seen.The signal s ( T ) = exp( − χ ( T )) is determined by thefrequency integral χ ( T ) := Z ∞ S ( ω ) ω F ( ωT ) dω (7)where F ( ωT ) is the filter function. For pulses of duration τ ( j ) centered at instants δ j T it is given by F ( z ) = (cid:12)(cid:12)(cid:12) − N +1 e − iz + 2 N X j =1 e izδ j cos (cid:0) zτ ( j ) T (cid:1)(cid:12)(cid:12)(cid:12) , (8)where we use z := ωT for brevity. This equation is validif the coupling between qubit and bath is effectively zeroduring the pulse. For artifical noise this can be realizedexperimentally [13] while for generic systems the pulsedesign has to approximate this situation [25, 27]. Clearly,for larger and larger T the influence of the finite pulsedurations τ ( j ) decreases more and more.The scaling of ∆ pF with T for UDD with ideal pulsesis also remarkable. For UDD, d / | b + − | + | b −− | where b + − and b −− depend on B , B z and on the initial densitymatrix ρ [36]. In particular, b −− ∝ T N +1) and its pref-actor is even in B z while b + − ∝ T ( N +1) with a prefactorodd in B z . If both are present one has the generic result d = O ( T N +1 ). But if the Hamiltonian is symmetric un-der global spin flip σ z ↔ − σ z , realized, e.g., by a π rota-tion about total σ x , it follows that b + − = − b + − = 0 due to its oddness in B z such that we obtain d = O ( T N +1) )which is better than generically expected. Hence on theone hand, the generic behavior of dynamic decouplingcan only be seen for systems without symmetry. On theother hand, we stick here to the Hamiltonian (1) becauseit is of the kind occurring mostly in experiment [2, 29, 41].Fig. 3(c) with α = 10 and Fig. 4(a) with α = 100 differin the rapidity of the bath dynamics which is faster forlarger α . Clearly, the decoherence sets in earlier if thebath is faster because the switching by the pulses is rela-tively slower. This is no contradiction to the basic idea ofmotional narrowing stating that a very fast bath implieslonger coherence times because the fast bath dynamicsreduces its influence on the qubit due to averaging. Butprevious results, e.g., Fig. 3 in Ref. 18, show that for thiseffect to take place α should exceed 10 .We do not consider data for smaller α / α > T where ∆ pF is dominated by the properties ofthe pulses. Naturally, this effect is most prominent forthe uncorrected rectangular pulses of 0 th order. In Fig.3(a) the distance d is significantly larger for pulses offinite width (symbols) than for the ideal ones (lines).The RUDD sequence performs worse than the other se-quences. This is not surprising since it is based on theassumption that the pulse is designed such that there isnone or no significant coupling between qubit and bathduring the pulse. A rectangular pulse realizes this as-sumption only in order τ ∗ .Hence it is clear that the level for ∆ pF which can bereached for small values of T is lower for the 1 st or-der pulses (panel (b)) and even lower for the 2 nd orderpulses (panel (c)). This fact illustrates nicely that theoptimization of pulses is indeed an important ingredi-ent in enhancing the performance of dynamic decoupling[21, 23, 25, 35].The key observation is that the τ j RUDD becomes thebest performing for j = 2. For j = 0 and j = 1 the τ j UDD sequence turned out to be more advantageous.We conclude that the pulses need to be sufficiently welldesigned in order that the underlying idea of the RUDDsequence [27] really pays. In Fig. 3(c) the gain usingRUDD instead of UDD is about two orders of magnitude.Such improvements are to be expected in the regimewhere the performance of the sequences is dominated bythe pulse errors.We emphasize that the fact that RUDD performs bet-ter than UDD or any other generic sequence of pulsesof constant duration is quite remarkable because most ofthe pulses in the RUDD sequence are much longer than τ ∗ . The sum T p of the lengths of all N pulses is T p = N τ ∗ for a generic sequence while it is T p = τ ∗ cot( π/ (2( N + 1)) / sin( π/ ( N + 1)) (9a) ≈ τ ∗ N + 1) /π for N large (9b)for the RUDD sequence according to Eqs. (4,5). One mayprefer to consider the total energy necessary to realizethe sequence [4]. The energy required for a given pulseis proportional to 1 /τ . Hence the total energy E p isgiven for the UDD sequence by E p = AN/τ ∗ where A is a constant depending on the shape of the pulse. Notethe linear divergence in N . In contrast, for the RUDDsequence one obtains E p = A sin( π/ ( N + 1)) τ ∗ N X j =1 πj/ ( N + 1)) (10a) ≈ (2 A/τ ∗ ) ln [2( N + 1) /π ] for N large (10b)which diverges only logarithmically in N . Thus, givena minimum pulse duration τ ∗ it is much less costly inenergy to reach long coherence times by applying RUDDthan by any generic sequence with pulses of constant τ ∗ .In view of the above observations, it remains to clarifywhy the RUDD works better than the other sequences,but only for higher order pulses. According to the ana-lytic foundation of RUDD [27], its advantage over othersequences with shaped pulses consists in the vanishing ofmixed terms in T and τ ∗ . For instance, an ideal UDD N scales generically like T N +1 and the τ j UDD N of N finite-width pulses certainly has errors scaling like T N +1 and( τ ∗ ) j +1 . But one cannot exclude the occurrence of termssuch as T τ ∗ , T τ ∗ , or T ( τ ∗ ) . They result from the in-terplay between the finite duration of the pulses and thesequence. It is crucial that this is different for τ j RUDD N .There the finite duration is fully taken into account inthe design of the sequence [27]. Hence the errors of the τ j RUDD N are of the order T N +1 and ( τ ∗ ) j +1 ; the lowestmixed terms are T N +1 τ ∗ and T ( τ ∗ ) j +1 .The above argument lays the foundation why theRUDD outperforms other sequences. To illustrate the ar-gument we plot the dependence of ∆ pF on τ ∗ for varioussequences of finite-width pulses in Fig. 5 for the centralspin model at M = 8. Results for the spin chain model(not shown) look very much the same except for a rescal-ing of T . In panel (a) all sequences behave similarly; thedependence on τ ∗ is linear, and the RUDD behaves worst.This fact is attributed to the larger average length of thepulses. Note that in the regime depicted the distance∆ pF is still fully dominated by the pulse errors.In panel (b) we can nicely see the crossover from theregime where the pulse error dominates (straight linescorresponding to ( τ ∗ ) ) to the saturation levels corre-sponding to the errors of the ideal CPMG and CDD se-quence. The errors of the ideal UDD sequence is muchlower so that its saturation level cannot be seen. Still the τ RUDD behaves worse than the τ UDD. -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 ∆ p F τ CPMG τ CDD τ UDD τ RUDD (a) central spinM=8 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 ∆ p F τ CPMG τ CDD τ UDD τ RUDD (b) central spinM=8 -5 -4 -3 τ * [1/ λ ] -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 τ CPMG τ CDD τ UDD -5 -4 -3 τ * [1/ λ ] -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 ∆ p F τ RUDD with 2 π pulse τ RUDD w/o 2 π pulse ∆ pF ∝ (τ * ) ∆ pF ∝ (τ * ) M=8central spin(c)
FIG. 5: (Color online) Distance ∆ pF vs. the shortest pulseduration τ ∗ for various sequences at α = 10 with T = 0 . /λ ;the panels correspond to pulses of finite width of differentorder as in Fig. 3. In panel (c) we again see the crossover from pulse errorsto sequence errors on τ ∗ →
0. Interestingly, the RUDDbehaves better than the UDD in that the pulse errorsdecrease expectedly faster ∆ pF,RUDD ∝ ( τ ∗ ) comparedto ∆ pF,UDD ∝ ( τ ∗ ) . We stress that the latter scaling isno contradiction to the pulse being second order becausean error T ( τ ∗ ) is not excluded. Fig. 5(c) establishes thatsuch mixed terms indeed deteriorate the performance ofunadapted sequences of finite-width pulses. This clarifiesthe behavior of RUDD relative to other sequences.For practical implementation, it is important to pointout that the behavior of τ RUDD for small τ ∗ is indepen-dent of whether or not we include the very short bound-ary 2 π pulses, cf. solid line and circles in Fig. 5(c). Thisis due to the shortness of these effective identity pulses.Last but not least, we find another regime of low val-ues of ∆ pF . This is the regime where the pulse lengthsreach their maximum value because the pulses touch oneanother. They are back to back. Quite unexpectedly, thefull RUDD including the boundary 2 π pulses again per-mits to obtain an extremely good suppression of decoher-ence. This regime is very interesting because it requiresonly very low pulse amplitudes and a small total energyfor the coherent control, cf. Eq. (10), due to the pulses ofmaximum length. Further studies of this relevant regimeare left to future research. Conclusions.
The analysis of sequences of finite-width pulses allows us to draw the following conclusions.They are derived from the data for the models studied,but we expect them to hold more generally.First, the use of higher order pulses generically impliesa significant improvement. Such pulses are designed suchthat they suppress the coupling to the bath to a highorder during their action [25]. Second, non-equidistantsequences such as UDD outperform or, in the worst case,perform the same as equidistant (CPMG) or concate-nated (CDD) sequences.Third, in the regime, where the pulse errors dominatethe suppression of decoherence is further enhanced byvarying the pulse durations along the sequence (RUDD)as suggested on analytic grounds [27]. This enhancementtakes only place for pulses of sufficient high order. Wefound that it is present for second order pulses. Thisestablishes RUDD as a promising concept and representsour central result.Fourth, an additional interesting asset of the RUDDis that the total energy required for the coherent controlby pulses increases only logarithmically with the numberof pulses – in contrast to all other sequences of unvariedpulses. Hence in particular long coherence times can berealized at low energy price.Fifth, surprisingly, we found an additional regimewhere the RUDD suppresses decoherence efficiently. Thisis the regime where the pulses are (almost) back-to-backapproaching continuous modulation [4]. Because in thisregime the pulses reach their maximum length the re-quired control energy is a minimum. Further research isrequired to study this promising regime in detail. ∗ Electronic address: [email protected][1] M. A. Nielsen and I. L. Chuang,
Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000).[2] U. Haeberlen,
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