Towards optimized suppression of dephasing in systems subject to pulse timing constraints

Thomas E. Hodgson,1 Lorenza Viola,2 and Irene D’Amico11Department of Physics, University of York, Heslington, York, YO10 5DD, United Kingdom

2Department of Physics and Astronomy, 6127 Wilder Laboratory, Dartmouth College, Hanover, New Hampshire 03755, USA(Dated: December 10, 2009)

We investigate the effectiveness of different dynamical decoupling protocols for storage of a single qubit inthe presence of a purely dephasing bosonic bath, with emphasis on comparing quantum coherence preservationunder uniform vs. non-uniform delay times between pulses. In the limit of instantaneous bit-flip pulses, this isaccomplished by establishing a new representation of the controlled qubit evolution, where the resulting deco-herence behaviour is directly expressed in terms of the freeevolution. Simple analytical expressions are givento approximate the long- and short- term coherence behaviour for both ohmic and supra-ohmic environments.We focus on systems with physical constraints on achievabletime delays, with emphasis on pure dephasing ofexcitonic qubits in quantum dots. Our analysis shows that little advantage of high-level decoupling schemesbased on concatenated or optimal design is to be expected if operational constraints prevent pulses to be appliedsufficiently fast. In such constrained scenarios, we demonstrate how simple modifications of repeated periodicecho protocols can offer significantly improved coherence preservation in realistic parameter regimes.

PACS numbers: 03.67.Lx,03.65.Yz,03.67.Pp,73.21.La

I. INTRODUCTION

The ability to effectively counteract decoherence processesin physical quantum information processing (QIP) devices isa fundamental prerequisite for taking advantage of the addedpower promised by quantum computation and quantum sim-ulation as compared to purely classical methods. Dynami-cal decoupling (DD) techniques for open quantum systems1,2

have been shown to be able to significantly suppress non-Markovian decoherence for storage times that can be verylong relative to the typical time scales associated with thedecoherence process itself. Over the last decade, the de-sign and characterization of viable DD schemes for realis-tic qubit devices has spurred an intense theoretical and ex-perimental effort, taking DD well beyond the original nu-clear magnetic resonance (NMR) setting3. While earlier DDschemes relied on the simple periodic repetition of instan-taneous pulses (so-called ‘bang-bang’ periodic DD, PDD2,and its closely-related time-symmetrized version, so-calledCarr-Purcell DD, CPDD3,4), recent theoretical investigationshave explored the benefits of more sophisticated control de-sign in a number of ways. In particular, this has led to de-vising recursive and randomized pulse sequences for genericdecoherence models on finite-dimensional quantum systems– so-called ‘concatenated’ DD (CDD5,6) and ‘randomized’DD7,8; to identifying ‘optimal’ protocols for a single qubitundergoing pure dephasing – most notably, the so-calledUhrig DD (UDD)9,10,11,12,13,14, and its extension to ‘locallyoptimized’15,16 DD sequences tailored to specific noise envi-ronments; and, most recently, to combining the advantagesof concatenation and optimization for a single qubit exposedto arbitrary decoherence17,18,19. As a key common feature,these investigations highlight the sensitivity of DD perfor-mance to the details of the applied control path, and point tothe importance of carefully tuning the relative pulse delaysin order to boost the efficiency of the achievable decoherencesuppression20.

In view of the above rich scenario, assessing the perfor-

mance of different DD protocolsin specific qubit devicesand/or in the presence of specific control constraintsbecomesespecially important. Recently, the effectiveness of traditionalmulti-pulse spin-echo sequences based on PDD and CPDD, ascompared to ‘high-level’ protocols based on CDD and UDD,has been scrutinized in several control settings. In particu-lar, a number of theoretical studies have addressed suppres-sion of pure dephasing associated to spectral diffusion23 andhyperfine-induced decoherence24 from a quantum spin-bathfor an electron-spin qubit, as well as suppression of classical1/f phase noise in a superconducting qubit25,26. Experimen-tally, the performance of CDD protocols has been character-ized for an NMR spin qubit27, while optimal UDD implemen-tations have been reported for both a trapped ion qubit ex-posed to engineered classical phase noise15,16,28and, most re-cently, for electron spin qubits undergoing spin-bath decoher-ence in a malonic acid crystal29. These studies have demon-strated, in particular, how UDD can significantly outperformlow-level DD schemes provided that the noise spectrum has asharp high-frequency cutoff and sufficiently high pulse repe-tition rates may be afforded.

Amongst prospective solid-state QIP platforms, excitonqubits in self-assembled quantum dots (QDs) have likewisereceived vast attention in recent years30,31: due to the cou-pling to photons, excitons can be driven all-optically on sub-picosecond time scales30. Excitonic implementations also al-low the flexibility of designing hybrid solid state-flying qubitschemes32,33. Pure dephasing turns out to be the dominantfactor limiting the coherence lifetime in such qubit devices,where strong coupling with phonon modes of the host crystalresult in typical decoherence (T2) time scales of a few pico-seconds34. We have previously shown in Ref. 35 that, remark-ably, PDD allows for substantial exciton coherence recoveryin experimentally relevant parameter regimes (up to90% re-covery over∼ 10 ps at room temperature), the control per-formance being especially enhanced for QD shapes and biasfields optimized for quantum computing architectures. Ourgoal in this paper is to quantitatively assess to what extent

more elaborated DD schemes – in particular, sequences em-ploying non-uniform pulse timings – can improve beyond thesimplest PDD setting when alower bound on the achievablecontrol time scale(minimum pulse separation) is present.

We find that in the presence of such a timing limitation,simple protocols such as PDD or CPDD may outperform high-level sequences based on CDD/UDD. Interestingly, on the onehand this reinforces similar conclusions drawn in Ref. 26 forclassical dephasing in superconducting qubits. On the otherhand, we additionally show how it is possible to engineera suitable ‘preparatory’ sequence that enhances the perfor-mance of a subsequent PDD pulse train. In the process, wetake advantage of the exact solvability of a purely dephas-ing model in the presence of instantaneous pulses to obtainan exact representation of the controlled dynamics in termsofthe free evolution. This allows rigorous results on the long-time asymptotic decoherence behaviour to be established forgeneric noise spectral densities, by allowing in particular acomparison between ohmic and supra-ohmic environments.Furthermore, our work provides a first explicit analysis ofCDD performance in the presence of a quantum bosonic bath.From a practical standpoint, our results suggest that simpleDD protocols may remain a method of choice if significanttiming constraints are in place, and that incorporating suchconstraints from the outset is necessary before further opti-mization can show its benefits. While our numerical resultsare tailored to excitons in QDs, we expect the above conclu-sions to be relevant for other constrained qubit devices.

II. SINGLE-QUBIT DEPHASING DYNAMICS

We consider the pure dephasing dynamics of a single qubitcoupled to a non-interacting bath of harmonic oscillators.TheHamiltonian of such a system may be written in the form

2σz + ~

ωjb†jbj

(g∗j b†j + gjbj)[(1 − α)σ0 + ασz ] (1)

≡ H0 + ~

(g∗j b†j + gjbj)[(1 − α)σ0 + ασz ], (2)

whereE gives the energy difference between the qubit’s lev-els, b†j andbj are canonical creation and annihilation opera-tors of the oscillator modej, andgj describes the couplingbetween the qubit and thej-th bath mode. In the above ex-pression forH , the parameterα accounts for the possibilitythat either both or only one of the spin (or pseudo-spin) qubitcomputational levels effectively couple to the bath:α = 1 cor-responds to the standard purely-dephasing spin-boson model,whereas ifα = 1/2, only theσz = +1 eigenstate couples tothe bath. Specifically, for an excitonic qubit, the logical statesare represented by the presence or absence a single (ground-state) exciton in the QD30, andE is the energy relative to thecrystal ground state.

As time evolves, the qubit becomes entangled with the en-

vironment and the off-diagonal elements of the qubit densitymatrix evaluated at timet in the interaction picture with re-spect toH0 read1,34

ρ01(t) = ρ∗10(t) = ρ01(t = 0)e−Γ(t), (3)

Γ(t) ≡ Γ0(t) (4)

= (2α)2∫ ∞

dωI(ω)

ω2coth

[1 − cos(ωt)],

whereT is the temperature,kB the Boltzmann’s constant, and

I(ω) =∑

δ(ω − ωj)|gj |2 (5)

is the spectral density function characterizing the interactionof the qubit with the oscillator bath. For a supra-ohmic envi-

ronment,I(ω)ω→0∼ ω3, as opposed, for instance, to an ohmic

reservoir whereI(ω)ω→0∼ ω. Likewise, the high-frequency

behaviour is characterized by a frequency cut-offωc, for in-stance, for excitons one can assume thatI(ω)

ω→∞∼ e−ω2/ω2

As it turns out, the decoherence of the qubit in the pres-ence of anarbitrary sequence of bang-bang pulses, each ef-fecting an instantaneousπ rotation, can still be exactly de-scribed by Eq. (3), provided a modified decoherence func-tion is used2,10,36. Consider an arbitrary storage timet, dur-ing which a total numbers of pulses is applied, at instants{t1, . . . , tn, . . . , ts}, with 0 < t1 < t2 < . . . ts < t. By usingthe theory developed by Uhrig in Refs. 10,11, we can define acontrolled coherence functionΓ(t) in the following way:

Γ(t) ≡

Γ0(t) t ≤ t1,Γn(t) tn < t ≤ tn+1, 0 < n < s,Γs(t) ts < t.

Here,Γ0(t) is given in Eq. (4) whereas for1 ≤ n ≤ s welet10

Γn(t) = (2α)2∫ ∞

2ω2coth

|yn(ωt)|2dω,

yn(z) = 1 + (−1)n+1eiz + 2

(−1)meizδm , z > 0,

with the n-th pulse being understood to occur at timetn =δnt, and0 < δ1 < . . . δn < . . . δs < 1. While the instanta-neous pulse assumption must be handled with care in general,we have discussed in Ref. 35 how it translates into reason-able physical constraints for an excitonic qubit coupled toaphononic bath.

We now proceed to directly relateΓn(t) to Γ0(t) for ar-bitrary n. Let us first rewrite the above coherence functionΓn(t) in a compact way as

Γn(t) =

∫ ∞

η(ω)|yn(ωt)|2dω, n ≥ 0, (7)

where we have defined

|y0(ωt)|2 ≡ |1 − eiωt|2, (8)

η(ω) = (2α)2I(ω)

2ω2coth

By relating|y1(ωt)|2 to |y0(ωt)|2 we can write

Γ1(t) = −Γ0(t) + 2Γ0(t1) + 2Γ0(t − t1). (10)

Upon continuing this iteration we find

Γ2(t) = −Γ1(t) + 2Γ1(t2) + 2Γ0(t − t1),

Γn(t) = −Γn−1(t) + 2Γn−1(tn) + 2Γ0(t − tn). (11)

Furthermore, by expressing|yn(ωt)|2 as a function of|y0(ωt)|2, we are able to write the entire evolution in the pres-ence of anarbitrary pulse sequence only in terms of the un-controlled evolution. Explicitly, we find:

Γn(t) = 2n

(−1)m+1Γ0(tm)

Γ0(tm − tj)(−1)m−1+j

(−1)m+nΓ0(t − tm) + (−1)nΓ0(t). (12)

The above equation is one of the main results of this paper. Byusing Eq. (12), it is, in particular, straightforward to seethat

Γn−1(tn) = limt→tn

Γn(t) = Γn(tn). (13)

This confirms that the functionΓ(t) as defined in Eqs. (6) iscontinuos at the (instantaneous) pulse timings, as expected onphysical grounds.

As a first example of the usefulness of this representation,we consider how two pulses may be used to increase theasymptotic coherence of a supra-ohmic system, in which thefree dephasing dynamics saturates in the long-time limit toa finite value34,37 Γ0(∞) > 0. Taking thet → ∞ limit inEq. (10) or, equivalently, lettingn = 1 in Eq. (12), yieldsΓ1(∞) = 2Γ0(t1) + Γ0(∞). SinceΓ0(t) ≥ 0 for all t, thisshows how a single pulse cannot decrease the asymptotic de-coherence level. However, after two pulses we have

Γ2(t) = Γ0(t) − 2Γ0(t − t1) − 2Γ0(t2)

+ 2Γ0(t1) + 4Γ0(t2 − t1) + 2Γ0(t − t2). (14)

Therefore,

Γ2(∞) = Γ0(∞)− 2Γ0(t2)+2Γ0(t1)+4Γ0(t2 − t1), (15)

andt1 andt2 can be chosen to decrease the asymptotic deco-

Γ ( 8)2

Γ ( 8)0

Γ ( 8)1

0 0 1 2 3 4 5 6

FIG. 1: Comparison betweenΓ0(t), Γ1(t), andΓ2(t) for an excitonqubit at T = 77 K, as computed from Eq. (6). Pulse times aret1 = 0.2 ps andt2 = 0.31 ps.

herence provided that

2Γ0(t2) − 2Γ0(t1) > 4Γ0(t2 − t1). (16)

In Fig. 1, we plotΓ(t) for an exciton qubit coupled to phononmodes and subject to two control pulses att1 = 0.2 ps andt2 = 0.31 ps. For comparison, we also plot the evolutionunder a single control pulse att1 = 0.2 ps and the free evo-lution Γ0(t). As one can see, Eq. (16) can indeed be satis-fied. Numerical results showing how a few pulses can increasethe asymptotic coherence have been reported for excitonic de-phasing in Ref. 38.

For the case of Fig. 1, as well as for all the numerical ex-amples in this paper, we consider (unless otherwise stated)an exciton qubit tightly confined within a GaAs QD at 77K. The QD potentials are modeled as parabolic in all threedimensions, with confinement energies in thez-direction of~ωe = 505 meV and~ωh = 100 meV, while~ωe = 30 meVand~ωh = 24 meV in the in-plane directions30,39. The sub-scripte/h indicates electron/hole, respectively. For this exci-ton, in the absence of control most of the coherence is lost af-ter a few picoseconds39. Having this specific system in mind,we shall setα = 1/2 henceforth in our numerical calcula-tions, and plot the quantity| exp(−Γn(t))|2, which is directlyproportional to the square modulus of the measured opticalpolarizationP(t).

As discussed in detail in Ref. 35, the spectral density of thissystem is given by

I(ω) = Ie(ω) + Ih(ω) + Ieh(ω), (17)

where the indicese/h/eh correspond to single particle spec-tral densities of the electron and the hole, and to the electron-hole inteference term respectively, and

Ie/h/eh(ω) =∑

Fe/h/ehi (ω) exp

−ω2

ω2ci,e/h/eh

. (18)

Here,i labels different phonon modes, whereasFe/h/ehi (ω)

is a mode-dependent function for whichFe/h/ehi (ω)

ω→0≤ ω3.

The spectral density may be further approximated as

I(ω) ≈ Fω3 exp(

−ω2

, (19)

where the parametersF andωc are determined by fitting acurve of the form Eq. (19) to the actual exciton spectral den-sity. For the particular exciton parameters listed above, thisyieldsF = 1.14 × 10−26s and~ωc = 2meV.

III. PERIODIC DD: PERFORMANCE AND EXACTASYMPTOTIC PROPERTIES

For a Hamiltonian as in Eq. (1), a DD cycle consisting oftwo uniformly spaced rotations byπ about thex axis,

X∆tX∆t, (20)

where time ordering is understood from right to left, removesthe interaction between the qubit and the boson bath2,5 to thelowest (perturbative) order inωcTc, with Tc = 2∆t. The sim-plest DD protocol, PDD, is obtained by iterating the abovecontrol cycle in time.

Fig. 2 compares the free evolution with the PDD-controlleddephasing for the exciton qubit under examination, computedfrom the exact expressions given in Sec II. Sequences withthree different pulse delays are shown,∆t = 0.1 ps, ∆t =0.2 ps, and∆t = 0.3 ps, respectively. For the exciton qubit,two conditions determine a suitable range of∆t for effectivePDD: i) On the one hand, it is necessary that the control timescaleTc be sufficiently short with respect to the (shortest) cor-relation time of the decoherence dynamics, which means inthis case2∆t . τc = 2π/ωc. Physically, this can also beinterpreted by requiring that the characteristic frequency in-troduced by the periodic control,

ωres =π

be significantly higher than the spectral cut-off fequency it-self, ωres & ωc, in such a way that the DD-renormalizedspectral density function,I(ω) tan2(ω∆t/2), is effectively‘up-shifted’ beyond the bath cutoff12,19,35,41. ii) On the otherhand, the existence of a lower bound on the pulse durationimplies a lower bound on the separation∆t in order for theinstantaneous-pulse description to be accurate. As discussedin Ref. 35, this means∆t & 0.1 ps for semiconductor self-assembled QDs of interest for QIP.

The values of∆t used in Fig. 2, are consistent with boththese conditions. It can be seen that coherence decays untilthefirst bit-flip occurs, after which it rises, reaches a local maxi-mum before decohering once again – with this pattern repeat-ing between every two bit-flips. It can also be seen that DDrecovers most of the dephasing, that is,exp(−Γ(t)) is muchcloser to unity than in the uncontrolled evolution, which fallsrapidly before saturating toexp(−Γ0(∞)). After the first fewinitial pulses, the dephasing enters a phase in which the be-

|e−Γ(t

t (ps)

t=0.1ps∆

t=0.3ps∆

∆ t=0.2ps

Free evolution

0 2 4 6 8 10

FIG. 2: | exp(−Γ(t))|2 for the exciton qubit in the presence of PDDwith ∆t = 0.1 ps,∆t = 0.2 ps,∆t = 0.3 ps, compared with thefree evolution determined byΓ0(t).

haviour after the(n+1)-th pulse is approximately the same asthe one after then-th pulse. For∆t = 0.1 ps and∆t = 0.2 ps,the average dephasingover each cyclein this ‘steady-state’phase is very small, leading to a practical ‘freezing’ of theaverage decoherence over a period much longer than the es-timated (sub-picosecond) gating times30. For ∆t = 0.3 ps,however, the increase of decoherence due to this average de-phasing with time is more noticeable, leading to worse DDperformance overall. It can also be seen that, to minimize theeffects of dephasing, any readout on the qubit should be madehalf-way between two control pulses. As it is well known inNMR, this motivates a proper choice of the observation win-dow, which underlies the Carr-Purcell (CP) sequence4 and isalso discussed in Ref. 42 in the spin-boson context.

A. Long-time dynamics: Ohmic versus supraohmic behaviour

A main advantage of the exact representation establishedin Eq. (12) is that it allows detailed quantitative insight onthe controlled dephasing behaviour to be gained. In particu-lar, we focus on long-time coherence properties, which havealso received recent attention in view of control-dependent‘saturation’ effects observed in the context of spin-bathdecoherence43 (see also Ref. 44). We start by quantifyinghow the decoherence function in the presence ofn pulses dif-fers between two consecutive control times. Let

∆Γn ≡ Γn(tn+1) − Γn−1(tn). (21)

By using Eq. (12) we obtain:

∆Γn = (−1)n[Γ0(tn+1) − Γ0(tn)] +

Γ0(tn+1 − tj)(−1)n+j

n−1∑

Γ0(tn − tj)(−1)j+n. (22)

∆Γn

∆Γ 8

∆Γ 8 3.775x10−4

3.524x10−3

=0.3ps

=0.25ps=

−0.04

−0.02

0 5 10 15 20 25 30 35 40

−0.04

−0.02

FIG. 3: Differential dephasing function,∆ΓPDDn , for the exciton qubit

under examination in the presence of PDD with∆t = 0.3 ps (top)and∆t = 0.25 ps (bottom), calculated from Eq. (22). The dot-ted lines show, in each case, the limiting value∆Γ∞ given by Eq.(23). Notice that forn < nsat, wherensat ∼ 15, the sign of∆ΓPDD

oscillates, in agreement with Eqs. (B5) and (B7).

Let now ∆ΓPDDn denote the above ‘differential dephasing

function,’ Eq. (21), specialized to a PDD protocol. Then, asshowed in Appendix A, the following asymptotic result holdsfor anarbitrary dephasing environment:

∆Γ∞ ≡ limn→∞

∆ΓPDDn = 8ωresη (ωres) . (23)

Interestingly, Eq. (23) can be used to describe how the de-phasing function changes betweenany two instants separatedby ∆t, for large enought. That is, consider

∆ΓPDDn (t̃) ≡ Γn+1(t̃ + tn+1) − Γn(t̃ + tn), (24)

where0 ≤ t̃ ≤ ∆t, tn = n∆t. By using Eq. (13) we canverify that ∆ΓPDD

n (0) = ∆ΓPDDn . Then one may also prove

(see Appendix B for detail) that

∆ΓPDDn (t̃)

n>nsat≈ ∆Γ∞, (25)

wherensat ≡ tsat/∆t is a sufficiently large integer defined inthe same Appendix. Eq. (25) shows that the dephasing in-crement becomesindependent ofn and t̃ for t > tsat, thatis, dephasing asymptotically enters a periodic oscillation ‘inphase’ with the PDD sequence. Thus,∆Γ∞ in Eq. (23) maybe used to describe the difference in dephasing between anytwo times separated by∆t – in particular, between consecu-tive coherence maxima which fort > tsat occur at̃t ≈ ∆t/2.For a supra-ohmic environment as in the exciton qubit, theconvergence of∆ΓPDD

n to ∆Γ∞, Eq. (23), is very fast. This isillustrated in Fig. 3 for two representative values of∆t.

Because∆Γ∞ in Eq. (23) is non-zero as long as∆t is fi-nite, we can infer thatΓn diverges for fixed∆t asn → ∞.

While this in principle implies a decay ofexp(−Γ(t)) to zerounder the PDD, details of the spectral density function (in-cluding the nature of the coupling spectrum and the form ofspectral cutoff) become essential to characterize different dy-namical regimes of interest. In what follows, we illustratethese features by contrasting ohmic and supraohmic dephas-ing environments, and by consideringstroboscopicsampling,tn = 2n∆t, in which case explicit analytic expressions for thePDD ‘filter function’ |y2n(2nω∆t)|2 are available. Specifi-cally, upon combining Eq. (11b) of Ref. 10 with Eq. (13) re-covers the well-known result1,11,42:

Γ2n(2n∆t) =

∫ ∞

4η(ω) sin2 (ωn∆t) tan2(ω∆t

In general, we expect two dominant contributions to theabove integral: the one from small values ofω, whereη(ω)is not small, and the one from the region of the resonance,ω ≈ ωres, where|y2n(ωt)|2 may be large. First, note that forboth a ohmic and supra-ohmic spectral density, the contribu-tions from the small-ω region saturate to afinite value withtime. For the ohmic case, this is true irrespective of the factthat the free dephasing dynamics doesnot exhibit a similarlong-time saturation. This behavior is due to the control termtan2(ω∆t/2), which increases the rate at which the integrandgoes to zero asω → 0. Second, the contribution from theω ≈ ωres region is more or less relevant depending on the formof the spectral cutoff. Clearly, such ‘resonating’ contributionsdo not pose a problem in the limiting situation of an arbitrarily‘hard’ spectral cutoff of the formΘ(ω − ωc) (Θ( ) denotingthe step function), since, as remarked earlier,ωres > ωc in agood DD limit. For a smooth (‘soft’) spectral cutoff, the res-onating contribution increases with time and will ultimatelybe responsible for the divergence ofΓ2n(2n∆t) asn → ∞.In fact,∆Γ∞ corresponds precisely to such a frequency range.As shown by Eq. (25), we can approximate∆Γn ≈ ∆Γ∞ fort > tsat: since at such long times, the contributions to Eq. (26)from smallω have saturated, dephasing is indeed dominatedfrom the region aroundωres. Thus, for both ohmic and supra-ohmic systems under PDD, the coherence will eventually de-cay to zero for large enough times and soft cutoffs.

The above considerations are illustrated in Fig. 4, where weplot exact results calculated from Eq. (26) for a representativeohmic spectral density with an exponential cutoff1:

Ia(ω) = Fω exp(

, (27)

In order to highlight the different contributions to the overalldephasing function, we also explicitly compute and plot thefollowing quantities: (i) (dotted line)

Γsmω(2n∆t) =

∫ ωres/2

η(ω)|y2n(ω2n∆t)|2dω, (28)

106 108

10000 100 1 0.01

FIG. 4: Dephasing behavior for an ohmic spectral density with expo-nential cutoff as in Eq. (27), withF = 0.5, α = 1/2, ∆t = 0.0015,ωc = 100 andT = 100ωc, in units where~ = kB = 1. While stro-boscopic sampling is implied, continuous interpolating lines are usedfor clarity. a) Full decoherence function,exp(−Γ(tn)), Eq. (26)(solid line); low-frequency contribution,exp(−Γsmω(t)), Eq. (28)(dotted line); resonanting contribution,exp(−Γres(t)), Eq. (29)(dashed line) versus rescaled timeωct. b) Comparison betweenexp(−Γres(2n∆t)) (Eq. (29)) (points) andexp(−∆Γ∞t/∆t) (solidline).

which isolates the small-ω contributions, and (ii) (dashed line)

Γres(2n∆t) =

∫ 3ωres/2

ωres/2

η(ω)|y2n(ω2n∆t)|2dω, (29)

which isolates the contributions from theω ≈ ωres region.Three distinct regions may be identified: an initial drop in co-herence due to the low-frequency modes, until saturation ofEq. (28) occurs at aboutt = τc; a plateaux region where thecontributions from Eq. (29) are not important enough to causefurther decoherence; and a final decay of coherence to zerocaused by increasing contributions from theω ≈ ωres region.Fig. 4 also compares (bottom panel) the resonating contribu-tions calculated from Eq. (29) with the asymptotic predictionexp(−∆Γ∞t/∆t) (solid line), with∆Γ∞ = 4.507 × 10−7.The data confirm that∆Γ∞ does indeed arise from the res-onating contributions as expected, and that as long as the low-frequency contributions have saturated,∆Γ∞ may be usedto accurately describe dephasing under PDD in the long timelimit, that is,∆Γn ≈ ∆Γ∞, for t > tsat.

Additional insight may be gained by examining how theabove different regimes (initial decay, plateaux, final coher-ence decay) are affected by the harder or softer spectral cutofffunction. Beside the ohmic spectral density of Eq. (27), con-

106 108

1 100 10000 0.01

FIG. 5: Dephasing behavior for supraohmic spectral densities withdifferent cutoffs, Eqs. (30)-(31). Notice that nowF = 0.0001,while all other parameters are as in Fig. 4.exp(−Γ(t)) (solid line),exp(−Γsmω(t)) (dotted line), andexp(−Γres(t)) (dashed line) asa function of the rescaled timeωct for spectral densitiesIb (upperpanel), andIc (lower panel), respectively.

sider the following supra-ohmic spectral densities:

Ib(ω) = Fω3 exp(

, (30)

Ic(ω) = Fω3 exp(

−ω2

, (31)

where, in particular,Ic(ω) has a Gaussian tail, similar to theexcitonic qubit case. When comparingIb(ω) andIc(ω) (seeFig. 5), the harder cut-off due to the Gaussian tail stronglyreduces the value ofη(ωres), and hence greatly increases theduration of the plateaux regime. In fact, for the set of param-eter chosen, our numerics loose the necessary precision wellbefore the third regime sets on forIc(ω). The harder cut-off ofthe Gaussian case also decreasesΓsmω and, in turn, decreasesthe decoherence that occurs before the plateux.

B. Short time dynamics

In the previous section we analyzed the dephasing dynam-ics in the presence of PDD fort > tsat. Here, we focus ont < tsat. The long time regime is entered whenΓn+1 =Γn + ∆Γ∞, and for this to occur the coherence must oscillatein phase with the DD pulses. However, the natural response ofthe coherence after the first PDD pulse is instead to oscillatewith a period of2∆t (twice that of PDD pulses, recall Fig. 6).This follows from the fact that the first bit-flip occurs an in-

terval∆t after a maximum,Γ0(0), and for sufficiently small∆t, the dephasing function is roughly symmetrical about thecontrol pulse, so the coherence maximum following the firstpulse occurs att ≈ 2∆t. The PDD sequence quickly drivesthe coherence into phase with it, (see Fig. 6), but the first fewevenbit-flips in PDD occur near the coherence maxima, andthis worsens the performance of the control sequence. Thismay be seen by considering Eq. (11) at timet = tn + t̃, with0 < t̃ ≤ ∆t. By expanding the first and last terms to firstorder int̃, and considering thatΓ0(0) is a maximum, Eq. (11)rewrites as

Γn(t) ≈ −dΓn−1(tn)

dtt̃ + Γn−1(tn). (32)

The second term in the above equation is a constant, hencethere can be a coherence peak after then-th control pulsesonly if the derivativedΓn−1(tn)/dt > 0, as also pointed outin Ref. 42. In particular, again using Eq. (11), we can calculate

dΓn(tn)

Γn(tn + t̃) − Γn(tn)

t̃= −

dΓn−1(tn)

which shows that the larger the gradient ofΓn−1(tn), thefaster the coherence is retrieved immediately following thenth pulse. In particular, ifΓn−1(t) is locally flat at the time ofthen-th pulse,no coherence gaincan occur after that pulse.

We can see from Fig. 6 that as PDD drives the coherenceoscillations into phase with it,∆Γn has alternating sign forodd and evenn (cf. Eqs. (B5) and (B7)).∆Γn is initiallynegative for oddn and positive for evenn, while its magnitudedecreases until a timetav after which∆Γn becomes positivefor oddn and negative for evenn, before saturating to∆Γn =∆Γ∞. We see numerically thattav is independent of∆t, withtav ≈ 0.5 ps in our case. Furthermore, we can show fromEq. (A1) that if we consider the times at which the controlpulses occur (̃t = 0), then

∆ΓPDDn (0) = ∆ΓPDD

n−1(0) + (−1)n∆t2Γ′′0 (n∆t), (33)

d2Γ0(n∆t)

Γ0((n − 1)∆t) − 2Γ0(n∆t) + Γ0((n + 1)∆t)

∆t2.

From this expression we can understand the behaviour of thedephasing for PDD as the coherence oscillations are driveninto phase with the PDD pulses. Asn increases, the signof the last term in Eq. (33) alternates, and its magnitudedecreases asdΓ0(n∆t)/dt reaches a maximum before de-creasing and tending to zero (recall the behavior ofΓ0(t) inFig. 1). Thus, we can now rigorously definetav by the condi-tion d2Γ0(tav)/dt2 = 0, that is, when the gradient ofΓ0(t) ismaximum.

C. Practical considerations

Even if the qubit coherence eventually decays to zero un-der PDD in our excitonic system, for practical purposes we

t (ps)

0 0.5 1 1.5 2 2.5 0.94

FIG. 6: Short term dephasing of the exciton qubit under PDD, with∆t = 0.1 ps. The diamonds indicate the timing of the PDD pulses.

only need to suppress the dephasing for the qubit lifetime,T1. From the above discussion, we can estimate more pre-cisely how short∆t must be, in order for this to happen. Fort = n∆t + ∆t/2 > tsat, we can approximate the off-diagonaldensity matrix element at the maxima of coherence (where ameasurement would be made) as

ρ01[(n + 1/2)∆t] ≈ ρ01(0)e−Γnsat[(nsat+1/2)∆t]−(n−nsat)∆Γ∞ .(34)

Considering the long-time limit, if∆t is sufficiently smallandn ≫ nsat, we may neglect the coherence that is lost whilstt < tsat, and further approximate the dephasing as

ρ01[(n+1/2)∆t] ≈ ρ01(0)e−n∆Γ∞ ≈ ρ01(0)e−∆Γ∞

∆tt. (35)

Thus, in the long-time limit, we effectively have1/T eff2 =

∆Γ∞/∆t. A sufficient condition for the dephasing to be sup-pressed for the entire qubit lifetime is then

T eff2 =

∆Γ∞

& T1. (36)

Fig. 7 shows∆Γ∞/∆t as a function of∆t for the excitonqubit under consideration, for which1/T1 = 1 ns−1. It canbe seen that for∆t . 0.2 ps, PDD effectively suppressesdephasing for the entire lifetime. This is in excellent agree-ment with our previous results in Ref. 35, where we foundnumerically that∆t = 0.2 ps leads to efficient PDD, but, incomparison,∆t = 0.3 ps could only suppress the dephasingfor relatively short times.

IV. COMPARISON OF PDD WITH NON-UNIFORM DDSCHEMES

Having characterized the performance of the simplest DDscheme, where the control involves a single time scale∆t,we proceed to examine some of the high-level protocols men-tioned in the introduction, which involvenon-uniform pulsedelaysto a lesser or greater extent. While CPDD is both, his-

t (ps)∆

0 0.05 0.1 0.15 0.2 0.25 0.3

FIG. 7: Effective long-time coherence decay rate,1/T eff2 =

∆Γ∞/∆t, Eq. (36), as a function of∆t. The dashed line is thequbit inverse lifetime,1/T1 = 1 ns−1.

Sequence Pulse Timing

S0 Free(∆t)

S1 X∆tX∆t

S2 X[X∆tX∆t]X[X∆tX∆t] = ∆tX∆t∆tX∆t

S3 X[∆tX∆t∆tX∆t]X[∆tX∆t∆tX∆t]...

Sℓ XSℓ−1XSℓ−1

TABLE I: Concatenated pulses sequences for a purely dephasingsingle-qubit interaction. Time ordering is from right to left.

torically, the most established approach and, ultimately,one ofthe most effective, we defer its discussion until after the anal-ysis of CDD and UDD, since it turns out that for the supra-ohmic system at hand CPDD naturally suggests the optimiza-tion strategy that will be introduced in Sec. V.

A. Concatenated decoupling

Instead of repeating the basic control cycle given inEq. (20), CDD recursively concatenates it within itself. LetSℓ denote the sequence corresponding to theℓ-th level of con-catenation, as given in Table I.

For a qubit undergoing arbitrary decoherence, CDD witha ‘universal decoupling’ cycle given, for instance, by∆tX∆tZ∆tX∆tZ, has been shown5 to significantly outper-form PDD in the limit ∆t → 0. However, for purely de-phasing systems for which∆t has afinite lower limit, andfor single-axisprotocols constructed out of the basic cycle inEq. (20), the advantages of CDD are largely lost, and PDDmay be more efficient40. While different ways for comparingdifferent DD protocols can be considered5,24,26, we shall fo-cus here on comparing the efficiency of PDD and CDD at en-suring dephasing-protected storage of the exciton qubit for afixedtimeTstorage. In particular, for our calculations we chooseTstorage= 10 ps. This time is appropriate given the typical gat-ing time for exciton-based QIP, which is of the order of 1 ps30.

1. Single CDD cycle

GivenTstorageand the presence of a physical constraint on∆t, a first way to exploit CDD is to identify a minimum con-catenation level,ℓ∗, for which the length of the correspondingsequence,Tℓ∗ = 2ℓ∗∆t, exceedsTstorage. For a given∆t,increasingℓ beyond this point would not modify the resultsbecause the pulse timings overTstoragewould be unchanged.(see Table I). Figure 8 compares CDD and PDD for storageof an exciton qubit for different∆t. As expected from thegeneral analysis of Ref. 5, the efficiency of CDD increaseswith decreasing∆t. However, in the range of values underexploration, and with readout effected at the maxima of thecoherence curve, CDD is found to be more efficient than PDDonly if ∆t . 0.036 ps. The latter time scale is substantiallysmaller than physically allowed in our system.

We can understand the possible advantage of CDD by com-paring it with the long- and short-time behaviour of PDD(Secs. III A and III B respectively). Eq. (34) shows thatthe long-time performance of the protocol depends on∆Γ∞,andΓnsat[(nsat + 1/2)∆t]. For very small∆t (hence small∆Γ∞), PDD is not the most efficient scheme because it leadsto a value ofΓnsat[(nsat + 1/2)∆t] which may be greaterthan for other pulse sequences, due to the initially ‘out ofphase’ pulses. In the regime where CDD outperforms PDD(very small∆t), the contributions to dephasing from aroundω = ωres (see Sec. III A) are negligible for both sequencesover Tstorage, since fort > tsat both sequences preserve themaxima of coherence very close to the valueexp(−Γ(tmax

sat ))corresponding to the timetmax

sat of the first maximum that fol-lows tsat. The advantage of CDD (if any) comes from thedifferent behaviour of the dephasing over the first few controlpulses, that is, up tot = tsat. The timing of the pulses in theCDD sequence are similar to those of PDD, but withfewerpulses at the instants where the even pulses occur in PDD.These ‘missing’ pulses are those which would occur near themaxima of coherence in the initial stages of the sequence (seeinsets of Fig. 8), that is, the ones responsible for decreasingthe coherence maxima whilet < tsat in PDD (Sec. III B).These ‘missing’ pulses allow the dephasing to maintain itsnatural response frequency after the first bit-flip, and no lossof dephasing is needed to change the rate of the oscillationsof coherence. Therefore,ΓCDD(tmax

sat ) > ΓPDD(tmaxsat ), and for

t < Tstorage, Γ(t) ≈ Γ(tmaxsat ) for both PDD and CDD in the

limit of sufficiently small∆t.While the above explains why CDD may outperform PDD,

as soon as∆t is long enough such that∆Γ∞ is significantoverTstorage, PDD becomes the most efficient sequence. Theperiod of the coherence oscillations for CDD is twice the onefor the PDD sequence corresponding to the same∆t (see in-sets in Fig. 8), resulting in faster dephasing at long timest forCDD.

2. Periodic repetition of CDD cycles

A different use of CDD consists in truncating concatenationat a fixed level and periodically repeating the resulting ‘super-

|e−Γ(t

t (ps)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.99

0.9985

0.9995

0 0.1 0.2 0.3 0.4 0.5

0 1 2 3 4 5 6 7 8 9 10

0.97 0.975 0.98

0.985 0.99

0.995 1

0 0.2 0.4 0.6 0.8 1 1.2

FIG. 8: (Color online)| exp(−Γ(t))|2 for PDD (black line) com-pared with CDD (light line) for∆t = 0.016 ps (ℓ∗ = 10, top),∆t = 0.036 ps (ℓ∗ = 9, middle), and∆t = 0.055 ps (ℓ∗ = 8,bottom). Insets: Close-ups of the same evolutions at short times;the pulse timings are indicated as well, with crosses (CDD) and dia-monds (PDD).

cycle’, constructed from Table. I. For instance, truncation atℓ = 2 results in our purely dephasing case in a cycle of length4∆t, which is identical in structure to a CP cycle (see Sec.IV C), and whose periodic repetition we term PCDD2. Fora single qubit undergoing arbitrary decoherence, the corre-sponding PCDD2 protocol (constructed from a16-pulse basecycle) has been shown to be the best performer in suppressingthe effects of a quantum spin bath24,41,43.

Figure 9 shows a comparison of PDD and PCDDℓ proto-cols forℓ = 2, 3, for the shortest pulse separation compatiblewith the exciton qubit constraint,∆t = 0.1 ps. One can inferthat, for the∆t andTstoragevalues considered, PCDDℓ per-forms better (that is, displays higher coherence maxima) than

|e−Γ(t

t (ps)

0.84 0.88 0.92 0.96

0 0.5 1 1.5 2 2.5

0 1 2 3 4 5 6 7 8 9 10

FIG. 9: (Color online) Comparison of PDD (dark line) and PCDD(light line) protocols with∆t = 0.1 ps. Top: Second-level con-catenated cycle, PCDD2. Bottom: Third-level concatenated cycle,PCDD3. Inset: Zoom over the initial part of the time window withtimings of the pulse sequences explicitly indicated (diamonds forPDD and crosses for PCDD3).

PDD for ℓ = 2, but worse forℓ = 3. The difference betweenPCDD2 and PCDD3 may be understood as a consequence ofthe fact that in terms of a Magnus expansion21, concatenatedcycles with evenℓ are time-symmetric, thus cancel the inter-action with the phonon bath up to (at least) the second order.Over the time period shown, PCDD2 also outperforms stan-dard PDD (see Fig. 9, upper panel). However, the coherenceoscillations for PCDD2 occur over a period of2∆t since, af-ter the initial pulse, the sequence is equivalent to PDD withabase time interval of2∆t. Therefore, we expect PDD to bemore efficient for long storage times, as PCDD2 will yield alarger∆Γ∞ than a PDD sequence characterized by∆t, henceworse asymptotic performance.

B. Uhrig decoupling

We now assess the limitations of the optimal sequence pro-posed by Uhrig10 when significant restrictions on∆t are inplace. In UDD, consecutive pulses are spaced according to

δj = sin2( πj

2n + 2

, (37)

which implies, in particular, closely spaced pulses at the be-ginning and the end of the evolution period. Such a controlsequence strongly suppresses the dephasing for a storage time

|e−Γ(t

t (ps)

0 1 2 3 4 5 6 7 8 9 10 0

0 2 4 6 8 10

FIG. 10: | exp(−Γ(t))|2 for the exciton qubit in the presence ofUDD with a) 100 pulses, b)50 pulses, c)40 pulses, d)25 pulses,e)14 pulses corresponding to the best allowed sequence for the caseof the exciton qubit. For comparison, f) shows the free evolution.

of the order of10

tUDD ≈ (n + 1)τc

2π, (38)

whereτc denotes, as before (Sec. III), the relevant bath cor-relation time. As mentioned, with~ωc ≈ 2 meV, this corre-sponds toτc ≈ 2.06 ps. BeyondtUDD, the efficiency of UDDfalls rapidly. From Eq. (38), we find that for UDD to effi-ciently protect the exciton qubit overTstorage≈ 10 ps,n mustbe on the order of100. Fig. 10 shows the resulting UDD per-formance asn is decreased. It can be seen that asn . 100,the advantage of UDD is rapidly lost.

For our QD system, however, the main physical limita-tion is on the time delay between pulses. The shortest in-terval between control pulses in UDD,∆tUDD

min , is before thefirst pulse, and after the last pulse. From Eq. (37) we seethat such a sequence withn = 100 pulses over a period ofTstorage= 10 ps corresponds to∆tUDD

min = 2.4x10−3 ps, whichis roughlytwo orders of magnitudeless than that allowed bythe physical constraints for the exciton qubit in question.Evenfor a sequence consisting ofn = 40 pulses only (for whichthe efficiency is already poor as shown in Fig. 10, curve c)),∆tUDD

min = 1.5x10−2 ps, which is still an order of magnitudeshorter than allowed.

To respect the physical constraints, one may estimaate thatallowed UDD sequences should have a number of pulsesn . 14 within the intendedTstorage = 10 ps. Such a se-quence corresponds to curve e) in Fig. 10. It is then clearthat any UDD sequence compatible with our physical con-straints is outperformed by the best allowed PDD sequencewhich would preserve a coherence close to 1 for the same timewindow (see Fig. 2). Fig. 10 also shows that any constrainedUDD sequence performing like curve d) or worse would in-crease the dephasing compared with the free evolution, thatis,would result in decoherence acceleration. The reason for theshortfalls of UDD in our setting stems from the large spread ofthe control intervals(ti − ti−1). If we impose a lower boundon the minimum time interval, other intervals must take up

a considerable proportion of the total evolution time. Thisplaces a relatively large restriction on how many pulses maybe used within a given storage period, and eventually resultsin large amounts of dephasing during the long time delays inwhich no pulses occur.

C. Carr-Purcell decoupling

We now focus on analyzing more closely CPDD, which re-sults from the periodic repetition of a CP cycle of the form4

∆tCPX 2∆tCPX∆tCP. (39)

This also corresponds, as noted, to PCDD2 with ∆tCP = ∆t(cf. Table I). Specifically, we are interested in comparing aPDD sequence with a CPDD having thesame cycle time, Tc =2∆t, thus∆tCP = ∆t/2: though the corresponding pulse timeinterval may not be allowed by the physical constraints we areconsidering, this study will pave the way to be the analysis tobe developed in the next section.

Basically, CPDD may be viewed as a PDD protocol wherepulses are uniformly spaced by2∆tCP, except that the se-quence is displaced forward byt1 = ∆t/2, the time at whichthe first pulse is applied. As a consequence of the symmetryof the control propagator in Eq. (39) with respect to the cyclemid-point, it is well known3 that CPDD is a second-order pro-tocol as compared to standard (asymmetric) PDD, with lead-ing corrections of orderT 3

c . Using the exact representationestablished in Eq. (12), we will now assess the extent to whichCPDD improves over PDD for a purely dephasing system, andgain insight into asymptotic properties.

We begin by determining the dephasing half-way be-tween consecutive control pulses for the case of PDD. UsingEq. (12), we find

ΓPDDn

(−1)m+1Γ0(m∆t)

Γ0((m − j)∆t)(−1)m−1+j

(−1)m+nΓ0

2− m

+(−1)nΓ0

, (40)

which we may rewrite as

ΓPDDn

(−1)m+1Γ0(m∆t)

Γ0((m − j)∆t)(−1)m−1+j

(−1)k+1Γ0

k −1

+(−1)nΓ0

, (41)

wherek = n − m + 1. Similarly, by using Eq. (12), we mayalso determine the dephasing for CPDD with∆tCP = ∆t/2andti = (i − 1/2)∆t, that is,

ΓPDDn [(n + 1/2)∆t] = ΓCPDD

n [(n + 1/2)∆t]. (42)

This exactresult is illustrated in Fig. 11, where we plot thedephasing behaviour under PDD and CPDD for the excitonqubit with ∆t = 0.1 ps. As predicted by Eq. (42), the co-herence in the presence of each sequence is equal at timest = (n+1/2)∆t. Interestingly, fort > tsat, ΓPDD

n [(n+1/2)∆]are local maxima of coherence whereasΓCPDD

n [(n + 1/2)∆t]are local minima, proving CPDD to be much more efficientthan PDD provided that the time of the first pulse is allowedto bet1 = 0.05 ps.

|e−Γ(t

t (ps)

0 0.5 1 1.5 2 2.5 3 3.5 4

FIG. 11: Comparison of CPDD (dotted line) with∆tCP = 0.05 psand PDD (solid line) with∆t = 0.1 ps for the exciton system.

V. TOWARDS OPTIMIZED SEQUENCES IN THEPRESENCE OF PULSE TIMING CONSTRAINTS

Building on the understanding gained from the comparisonbetween different protocols in Sec. IV, we now specificallyaim to optimize DD performance for a bosonic dephasing en-vironment when pulses are subject to a minimum pulse-delayconstraint.

The basic observation is to note that if after an initial arbi-trary pulse sequence, PDD is turned on at a timetPDD, thenfor t > tPDD + tsat, we haveΓn+1(t + ∆t) − Γn(t) = ∆Γ∞

(recall Eq. (25)). This naturally suggests aninterpolated DDapproach, where an initial sequence is chosen to minimize

Γ(tmaxsat ), whilst transforming the oscillations of coherence into

phase with a PDD sequence to be turned on immediately af-terwards. Interestingly, a similar philosophy has been invokedto optimally merge deterministic and randomized DD meth-ods to enhance performance over the entire storage time45. Inour case, CPDD is indeed the simplest example of this inter-polation: as already noted, CPDD can be thought of as a PDDsequence applied attPDD = ∆t/2 + ∆t, following a prepara-tory sequence consisting of a single pulse att = ∆t/2.

Unfortunately, standard CPDD is not allowed in our sysyemdue to the physical constraint: the time interval betweenpulses in the initial sequence is smaller than the minimum al-lowed∆t which characterizes the subsequent PDD sequence.Simply using a CPDD sequence which does not break the timeconstraint is clearly not optimal. If the smallest allowed pulseinterval is∆tmin, then the best CPDD sequence consists ofperiodic repetitions of a CPDD cycle with∆tCP = ∆tmin,and the most efficient allowed PDD sequence is repetitions ofX∆tminX∆tmin. Since CPDD cancels the terms in the Mag-nus expansion up to to the second order, over the first fewrepetitions it performs much better than PDD, which only can-cels them up to the first order. However, for longer times theeffects due to the higher-order Magnus corrections accumu-late, and they turn out to do so more favorably for PDD. Thismanifests itself in a smaller∆Γ∞ for PDD than for the bestallowed CPDD protocol. As shown by Eqs. (B3) and (B6),the coherence oscillations are independent of the timing ofany pulses applied beforet − tsat. Therefore, CPDD can betreated as a PDD sequence with∆t = 2∆tCP for t > tsat.This justifies defining a∆Γ∞ for a CPDD sequence.

Physically, what is needed is a different initial sequence thatefficiently ‘engineers’ the transition of the coherence oscilla-tions – from the natural response frequency determined by thefirst bit-flip to the frequency of the following PPD sequence.To accomplish this, we propose to useCP cycles with vary-ing ∆tCP. That is, we define such an interpolated sequenceby letting theith cycle to be characterized by a pulse de-lay ∆tCP

i , and begin immediately after the previous cycle atti = ti−1 + 4∆tCP

i−1. The analysis of the resulting averag-ing properties may be carried out by adapting the derivationof Ref. 5 to the pure dephasing bosonic setting of Eq. (1).While the detail of the calculations are included in AppendixC, the result is that, similar to standard CP, the proposed DDsequence still cancels the terms in the Magnus expansion up tothe second order. Therefore, the interpolated scheme does notonly perform well for smallt, but also quickly results in pulsesuniformly separated by∆tmin – resulting in a small∆Γ∞,hence high performance for long storage times.

The simplest way to generate a good interpolated DD se-quence is to apply a CP cycle with∆tCP = ∆tmin, followedby periodic repetitions of one with∆tCP = ∆tmin/2. Thesequence is then given by

t1 = ∆tmin,

t2 = 3∆tmin,

t3 = 3∆tmin +3

2∆tmin,

ti = ti−1 + ∆tmin, i > 3. (43)

We compare this sequence with standard PDD with∆t =∆tmin in Fig. 12, upper panel. One clearly sees that the se-quence in Eq. (43) is more efficient.

By construction, the first two CP cycles in the above se-quence play the role of modifying the frequency of the de-phasing oscillations in such a way that they are brought inphase with the following repeated cycles. We can performthis process more smoothly by gradually reducing∆tCP from∆tmin to ∆tmin/2 over more than a control cycle. Though forvery small∆tmin the two cases would be equivalent, for sys-tems such as the exciton qubit where the time restrictions arerelatively severe, the smoother transition sequence may de-couple the qubit more efficiently. Such a modified sequencemay be implemented by applying CP cycles with decreasing∆tCP , that is,

∆tCPi =

∆tmin, i = 1

∆tmin − (i − 1)∆2, 1 < i ≤ iPDD,

∆tmin/2, i > iPDD,

whereiPDD = ∆tmin/(2∆2) and∆2 is an arbitrary time de-fined such thatiPDD is an integer. The greateriPDD, the longerthe time over which the the decreasing length cycles are ap-plied. Alternatively, we may describe the above sequence interms of the pulse times:

ti = ∆tmin + (i − 1)2∆tmin

−(i − 1)(i − 2)

2∆2, i <

∆tmin

∆2− 2,

ti = ti−1 + ∆tmin, i ≥∆tmin

∆2− 2. (45)

The above modified sequence is compared to PDD with∆t =∆tmin in Fig. 12, lower panel. As before, we see that DDwith varying CP cycles outperform PDD. Furthermore, it canbe seen that there is an improvement over the more abruptsequence described by Eq. (43).

In Sec. IV A 2, we showed that a constrained CPDD se-quence could outperform PDD over time scales of the orderof 10 ps (see top panel of Fig. 9 for PCDD2) even though,for longer times, the smaller∆Γ∞ for PDD would eventuallymake it more efficient than CPDD. In Fig. 13, we further com-pare CPDD with the interpolated sequence given in Eq. (44).The latter is found to be slightly more efficient than the bestallowed CPDD sequence over short time scales. Furthermore,because of the smaller∆Γ∞, as time progresses it will alsooutperform CPDD asymptotically. A main advantage of thesequence given in Eq. (44), however, is that it not only leadstohigher maxima than CPDD, but also, after the first few pulses,to a much smaller coherence oscillation amplitude. This re-flects the fact that the oscillation period has been tuned to theminimum allowed time interval∆tmin. In this respect, the per-formance of the sequence in Eq. (44) is more robust againstthe precise readout times, or, equivalently, a readout offset er-ror relative to the coherence maxima would not significantlyaffect the coherence recovered using this sequence.

|e−Γ(t

t (ps)

0 0.5 1 1.5 2 2.5

0 2 4 6 8 10

FIG. 12: (Color online) Comparison of PDD (dark line) with inter-polated DD sequences of varying CP cycles (light line), for∆tmin =0.1 ps. Top: Sequence given by Eq. (43). Bottom: Sequence givenby Eq. (44), with∆2 = 0.01 ps. Inset: Zoom over the initial part ofthe time window with timings of the pulse sequences explicitly indi-cated (diamonds for PDD and squares for the modified sequences).

VI. CONCLUSION AND OUTLOOK

We have investigated the ability of DD to inhibit decoher-ence of a single qubit coupled to a purely dephasing bosonicenvironment, by comparing the performance of low-level pe-riodic DD schemes based on uniform pulse separations tohigher-level non-uniform DD schemes.

For arbitrary spectral density functions characterizing dif-ferent dephasing environments, we have derived an exact rep-resentation of the controlled dynamics available for instanta-neous pulses, and exact relationships for the asymptotic dy-namics in the case of PDD. Building on these results, we haveshown that a main weakness of PDD is due to the oscilla-tion of coherence following the first bit-flip being out of phasewith the rest of the sequence. This has naturally suggested theapplication of a suitably engineered preparatory sequenceasa strategy to enhance DD efficiency, by bringing the coher-ence oscillations into phase with a subsequent PDD sequence.The resulting ‘interpolated’ DD protocols are found to be es-pecially efficient for physical systems where the mimimumtime interval between control pulses is strongly constrained.For such systems, DD protocols like concatenated or UhrigDD, which are designed to achieve peak performance whenthe asymptotic regime of arbitrarily small pulse separations is

|e−Γ(t

t (ps)

0 0.5 1 1.5 2 2.5 0.88

0 2 4 6 8 10

FIG. 13: (Color online) Comparison of the best allowed CPDD se-quence (light line) with a sequence of CP cycles of decreasing length(dark line) as described in Eq. (44) with∆2 = 0.01 ps and for∆tmin = 0.1 ps. Inset: Zoom over the initial part of the time windowwith timings of the pulse sequences explicitly indicated (squares forthe modified sequence and crosses for CPDD, respectively).

fully accessible, tend to largely lose their advantages.

For the excitonic dephasing environment of interest, in par-ticular, we have shown how a sequence of Carr-Purcell cycleswith suitably chosen (analytically generated) time delayspro-vides a very efficient DD protocol for realistic QD parame-ters and qubit storage times. Our process of constructing aDD sequence under which the coherence oscillates asymp-totically with the minimum period allowed by the physicalcontraints offers, as by-product, the advantage of asignif-icantly smaller coherence oscillation amplitude, relative toconstrained PCDD or UDD sequences. This makes the pro-posed interpolated sequence morerobustagainst readout.

While our analytically-designed interpolated DD proto-col might be compelling in its simplicity, identifying DDschemes that are guaranteed to yield optimal performancesubject to non-trivial timing constraints appears as an interest-ing control-theoretic problem for further investigation.Revis-iting the local numerical optimization approach recently pro-posed in Ref. 16 in aconstrained minimizationperspectivemight offer a concrete starting point in this respect. Like-wise, the investigation of dynamical error-control schemesbased on bounded-strength ‘Eulerian’ DD46, along with therecently proposed extension to decoherence-protected quan-tum gates47,48, might prove especially fruitful for excitonqubits, in view of the reduced control overheads associatedwith purely dephasing environments. Lastly, an interestinggeneral question is to what extent exact representations ofthecontrolled coherence dynamics in terms of the uncontrolledone may exist for an arbitrary purely dephasing error model,allowing, for instance, exact insight into the long-time con-trolled dynamics to be gained.

Acknowledgments

It is a pleasure to thank Kaveh Khodjasteh for a criticalreading of the manuscript. L.V. gratefully acknowledges par-tial support from the National Science Foundation throughGrants No. PHY-0555417 and No. PHY-0903727, and fromthe Department of Energy, Basic Energy Sciences, under Con-tract No. DE-AC02-07CH11358.

APPENDIX A: DERIVATION OF ∆Γ∞

For PDD,tn = n∆t, using this expression and recastingthe sums in Eq. (22) in terms ofk = n − j, we can write

∆ΓPDDn = (−1)nΓ0((n + 1)∆t) − 3Γ0(n∆t)(−1)n

n−1∑

Γ0(k∆t)(−1)k. (A1)

Using Eq. (7) forn = 0 and extending the sum to includek = 0, Eq. (A1) becomes

∆ΓPDDn = −4

∫ ∞

η(ω)dω + (−1)n

∫ ∞

6η(ω) cos(ωn∆t)dω

−(−1)n

∫ ∞

2η(ω) cos(ω(n + 1)∆t)dω

∫ ∞

η(ω)

n−1∑

cos(ωk∆t)(−1)kdω. (A2)

By using the relationship

(−1)k cos(kx) = −1

(−1)n cos(

2n+12 x

2 cos(

) , (A3)

we can rewrite the above equation as

∆ΓPDDn = 2

∫ ∞

η(ω){

(−1)n+1 cos[(n + 1)ωn∆t]

+ (−1)n+1 cos(nωn∆t)

+ 2(−1)n cos

2n+12 ∆tω

∆tω2

∫ ∞

η(ω){

(−1)n+1

× cos

2n + 1

2∆tω

∆tω

+(−1)n cos

2n+12 ∆tω

∆tω2

dω. (A4)

This finally can be rearranged as

∆ΓPDDn = 4

∫ ∞

η(ω) sin2(ω∆t

×(−1)n cos

n + 12

ω∆t2

) dω. (A5)

We now take the limit ofn → ∞, and note that

limn→∞

(−1)n cos(

n + 12

ω∆t2

∞∑

δ (ω − (ωres+ 2lωres)) . (A6)

If we assume thatη(ωres) ≫ η(ωres+ 2lωres) for l > 0, whichis true for sufficiently small∆t (that is,ωres ≫ ωc), we canneglect contributions froml > 0 and define

∆Γ∞ = limn→∞

∆ΓPDDn (A7)

∫ ∞

8dωδ(ω − ωres)η(ω)ωressin2(ω∆t

= 8η(ωres)ωres.

APPENDIX B: LONG-TIME LIMIT OF ∆ΓPDDn (t̃)

By using Eq. (12) and straightforward manipulations, wecan separate∆ΓPDD

n (t̃) from Eq. (24) into two parts,

∆ΓPDDn (t̃) = ∆ΓTI

n + ∆ΓTDn (t̃), (B1)

∆ΓTDn (t̃) = (−1)n

Γ0(tn + t̃) − Γ0(tn+1 + t̃)]

, (B2)

and the second term

∆ΓTIn = 2(−1)n

Γ0(tn+1) + 2

(−1)jΓ0(tn+1 − tj)

(B3)independent of̃t. By using

Γ0(t) = 2

∫ ∞

η(ω)[1 − cos(ωt)]dω, (B4)

we can rewrite∆ΓTDn (t̃) as

∆ΓTDn (t̃) = −4(−1)n

∫ ∞

η(ω) sin(∆tω

sin{[(

∆t + t̃]

dω. (B5)

The last term in the integrand above is fast oscillating for largen, so we will have that forn > nsat

∣∆ΓTDn (t̃)

∣ < ǫ, (B6)

whereǫ can be made arbitrarily small.

Let us now consider∆ΓTIn . By using Eq. (B4), the relation

Eq. (A3) and some tedious but straightforward manipulations,we can rewrite Eq. (B3) as

∆ΓTIn = −(−1)n4

∫ ∞

η(ω) cos[ω(n + 1)∆t]dω

∫ ∞

η(ω)(−1)n cos

2n+12 ∆tω

2 cos(

∆tω2

) dω. (B7)

Again, the integrand in the first term of the above equationis fast oscillating for largen, while the second term tends to∆Γ∞ for n → ∞ (see Eq. (A6). We can then write that forn > nsat,

|∆ΓTIn − ∆Γ∞| < ǫ. (B8)

By combining Eqs. (B6) and (B8), we finally obtain that forany t̃ andn > nsat,

∆ΓPDDn (t̃) ≈ ∆Γ∞. (B9)

We note that in the case of supraohmic environment, byusing thatΓ0(∞) ≡ limn→∞ Γ0(tn+1) = 2

∫ ∞

0 η(ω)dω isfinite, and using Eqs. (B2) and (B7), we can recast the condi-tions Eqs. (B6) and (B8) as

|Γ0(t > tsat) − Γ0(∞)| < ǫ, (B10)

wheretsat = nsat∆t. This emphasizes that condition (B9)applies for times at which thenatural evolution saturates toits long-term behavior.

APPENDIX C: AVERAGING PROPERTIES OFINTERPOLATED DD SCHEME

We begin by casting the QD Hamiltonian Eq. (1) (withα = 1/2) in the following form:

H1 = σz ⊗ Bz + σ0 ⊗ B0, (C1)

whereBz andB0 are operators acting on the phonon bath, andσ0 andσz denote the identity and the Pauli matrix acting onthe exciton qubit, respectively. This allows us to express theevolution in the presence of thei-th CP cycle by the propaga-tor

UCPi (4∆tCP

i ) = Uf(∆tCPi )XUf(2∆tCP

i )XUf (∆tCPi ),

whereUf(t) = exp(−tH1) represents free evolution for atime t. If we define

H2 ≡ −σz ⊗ Bz + σ0 ⊗ B0 = XH1X, (C2)

we can write the entire sequence propagator as a Magnus se-ries expansion21,

U(t) = exp

∞∑

Ai(t), (C3)

for which, in the limit of sufficiently fast control, we can onlyconsider the first two lowest-order terms in∆t5. Specifically(in units where~ = 1):

A1 = −i

dt1H(t1), (C4)

A2 = −1

∫ t1

dt2[H(t1), H(t2)], (C5)

whereH(t) = U †ctrl(t)HUctrl(t) is the time-dependent (piece-

wise constant, for instantaneous pulses) effective Hamilto-nian that describes the evolution under the control propagatorUctrl(t) resulting from the applied pulses2,3,5.

For the sequence of different CP cycles described in Sec.V, A1 is proportional to the identity operator, and hence doesnot contribute to dephasing. This is a simple consequenceof the qubit spending equal amounts of time in each of thecomputational basis states. More interestingly, we find

∫ tCPn

+4∆tCP

∫ t1

dt2[H(t1), H(t2)] =

∫ tCPi

+4∆tCP

∫ t1

dt2[H(t1), H(t2)]

∫ ti+∆tCP

∫ t1

∫ ti+3∆tCP

ti+∆tCP

∫ t1

∫ ti+4∆tCP

ti+3∆tCP

∫ t1

[H(t1), H(t2)]}

∫ ti+∆tCP

i−1∑

2∆tCPj [H1, H2] +

∫ ti+3∆tCP

ti+∆tCP

( i−1∑

2∆tCPj + ∆tCP

[H2, H1]

∫ ti+4∆tCP

ti+3∆tCP

2∆tCPj [H1, H2]

∆tCPi

i−1∑

2∆tCPj [H1, H2] + 2∆tCP

( i−1∑

2∆tCPj + ∆tCP

[H2, H1] + ∆tCPi

2∆tCPj [H1, H2]

again up to irrelevant pure-bath terms. This confirms the second-order cancellation claimed in the main text.

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