Jeffrey F Godsell

Tanushree Bala

Kevin M Ryan

Saibal Roy
email: saibal.roy@tyndall.ie

AC susceptibility Studies in capped Ni/Ni(OH) 2 Core/Shell nanoassemblies: Dual Peak Observations

In the present study, the AC susceptibility (χ` and χ``) variation with temperature (10-100K) for oleic acid capped Ni/Ni(OH) 2 core/shell nanoparticle assemblies are reported at frequencies varying from 0.1 Hz to 1000 Hz. Nanoparticle assemblies, with two average particle diameters of ~34 nm and ~14 nm were synthesised using a wet chemical synthesis approach. Two peaks in the AC susceptibility versus temperature curves are clearly discernable for each of the samples. The first, occurring at ~22K was attributed to the paramagnetic/antiferromagnetic transition of the Ni(OH) 2 present in the shell. The second higher temperature peak was attributed to the superparamagnetic blocking of the pure Ni situated at the core of the nanoparticles. The higher temperature peaks in both the χ` and χ`` curves were observed to increase with increasing frequency. Thus

the Néel and the blocking temperatures for such core-shell nano-assemblies were clearly identified from the AC analysis, where as they were not discernable (superimposed) even from very low DC (FC/ZFC) field measurements. Interparticle interactions within the assemblies were studied through the fitting of phenomenological laws to the experimental datasets. It is observed that even with an oleic acid capping layer, larger Ni/Ni(OH) 2 nanoparticles experience a greater degree of sub-capping layer oxidation thus producing lower magnetic interaction strengths.

I. Introduction

To date magnetic nanoparticles have been successfully synthesised (Godsell et al., 2010[START_REF] Li | Supercritical Fluid Synthesis of Magnetic Hexagonal Nanoplatelets of Magnetite[END_REF][START_REF] Banerjee | Magnetic properties of oxide-coated iron nanoparticles synthesized by electrodeposition[END_REF][START_REF] Donegan | Microwaveassisted synthesis of icosahedral nickel nanocrystals[END_REF][START_REF] Das | Interface controlled electrical and magnetic properties in Fe-Fe3O4-silica gel nanocomposites[END_REF] and used in a variety of applications such as in magnetic resonance imaging contrasting agents [START_REF] Huh | In Vivo Magnetic Resonance Detection of Cancer by Using Multifunctional Magnetic Nanocrystals[END_REF], ferro-fluids [START_REF] Gomes | Synthesis of Core-Shell Ferrite Nanoparticles for Ferrofluids: Chemical and Magnetic Analysis[END_REF], magnetic storage devices [START_REF] Speliotis | Magnetic recording beyond the first 100 Years[END_REF], catalytic applications [START_REF] Osaka | New Trends in Nanoparticles: Syntheses and Their Applications to Fuel Cells, Health Care, and Magnetic Storage[END_REF], hyperthermia treatment [START_REF] Jordan | Magnetic fluid hyperthermia (MFH): Cancer treatment with AC magnetic field induced excitation of biocompatible superparamagnetic nanoparticles[END_REF], drug delivery [START_REF] Mcgill | Magnetically Responsive Nanoparticles for Drug Delivery Applications Using Low Magnetic Field Strengths[END_REF] etc.

The effects of interparticle and surface interactions on magnetic properties has continued to be a central issue in the study of magnetic nanoparticle systems.

Interparticle interaction effects between magnetic nanoparticles have previously been analysed through the use of AC susceptibility measurements with the aid of the Néel-Arrhenius and the Vogel-Fulcher laws. In such studies, the frequency of the AC excitation field being applied to the nanoparticles is progressively incremented while the complex susceptibility is measured across a varying temperature range to allow for an exploration of the relaxation processes within the sample.

For the present study, a system of Ni/Ni(OH) 2 (core/shell), nanoparticles were analysed. For Ni nanoparticles synthesised within an aqueous environment, it was shown [START_REF] Bala | Block copolymer mediated stabilization of sub-5 nm superparamagnetic nickel nanoparticles in an aqueous medium[END_REF] that air drying post centrifugation could lead to the formation of an amount of Ni(OH) 2 within the samples. In spite of the presence of an oleic acid (OA) capping layer, a thin shell of Ni(OH) 2 was formed, to create a Ni/Ni(OH) 2 core/shell type structure. Such a structure allows for an investigation of how the strength of interactions between the particle cores is affected by the presence of a Ni(OH) 2 shell around each of the nanoparticles comprising the assemblage. The ability to slightly oxidise the surface of a Ni nanoparticle may be advantageous in certain applications such as in protein separation [START_REF] Lee | Ni/NiO Core/Shell Nanoparticles for Selective Binding and Magnetic Separation of Histidine-Tagged Proteins[END_REF], allowing for a tailoring of the sample's magnetic properties. Such effects as a control of magnetisation or the introduction of an exchange bias/pinning effect at lower temperatures are easily achievable with such a core/shell structure. Indeed it has been reported that an exchange bias effect in nanoparticles with antiferromagnetic shells can overcome the superparamagnetic limit, which holds potential interest for the magnetic data storage industry [START_REF] Skumryev | Beating the superparamagnetic limit with exchange bias[END_REF].The potentially interesting control of the magnetic properties allowed by this bilayer Ni/Ni(OH) 2 nanostructure were hence deemed worthy of further investigation as detailed in the subsequent sections.

II. Experiment

Details of the particle synthesis technique have previously been reported [START_REF] Bala | Block copolymer mediated stabilization of sub-5 nm superparamagnetic nickel nanoparticles in an aqueous medium[END_REF]. In brief, an aqueous mixture of 1 x 10 -2 M Ni(NO 3 ) 2 •6H 2 O was mixed with 1 x 10 -4 M oleic acid and 1 x 10 -2 M (sodium dodecyl sulphate) SDS. To produce sample 2 an amount of Pluronic block copolymer (P123) was added to keep the concentration at 0.3% in the solution, however no P123 was added to produce sample 1. The solutions were reduced using 0.035g of NaBH 4 in solid form. Post addition of the reducing agent to the solutions the liquid was seen to turn black almost immediately. The solutions were allowed to stand for 30 minutes to ensure the completion of the reduction reaction before repeatedly being centrifuged at 8000rpm for 15 minutes to separate the pellets. The pellets were then washed using deionised water, the resultant pellets were subsequently dispersed in water. The samples were next fully dried in an air environment to produce a dry powdered sample for magnetic analysis.

The reactions were seen to be perfectly reproducible with the as synthesised particles being reasonably monodisperse. XRD analysis showed the presence of both fcc Ni (PCPDF file no. 04-0850) and Ni(OH) 2 in the sample, figure 1. The presence of peaks at 33.8 and 51.5 ensured the hydroxide to be -Ni(OH) 2 [START_REF] Subbaiah | Characterisation of nickel hydroxide precipitated from solutions containing Ni2+ complexing agents[END_REF]. Figures 2 showed the TEM and HRTEM images of the images of samples 1.

A particle distribution analysis (inset, Figure 2A) revealed the size as ~34 nm in diameter. Ni (111) planes could be identified in the HRTEM image (Figure 2C) but no planes for Ni(OH) 2 were visible plausibly due to poor crystallinity of the thin hydroxide layers . A detailed XPS analysis [START_REF] Bala | Block copolymer mediated stabilization of sub-5 nm superparamagnetic nickel nanoparticles in an aqueous medium[END_REF] for the same sample demonstrated fairly good amounts of Ni(OH) 2 species. Considering the surface sensitivity of the XPS technique and combining the HRTEM and XPS results, the presence of the hydroxide layers on the surface of the nanoparticles was inferred.. While sample 1 displays particles ~34nm in diameter, sample 2 is seen to be initially bimodal with a small fraction of the particles ~34nm in size but the predominant portion of particles are ~14nm. With sample 2, repeated centrifugation was used to remove the ~34nm particles from the bimodal sample to leave predominantly ~14nm samples behind for magnetic analysis. For both samples, magnetic measurements were performed using a commercial superconducting quantum interference device, (Quantum Design -Model no. MPMS-XL5). For the AC susceptibility measurements, a peak drive field amplitude of 2.5Oe was applied for all frequencies, no DC offset magnetic field was applied during AC measurement.

III. Theory

With varying AC frequency, the percentage of magnetic nanoparticles which follow the applied field's oscillations will change, inducing an associated modification in the sample's magnetic response. For non-interacting nanoparticle systems, which in turn are subjected to a slowly oscillating magnetic field (h=h o cosω m t), the real and imaginary parts of the ac susceptibility's components are given by [START_REF] Singh | ac susceptibility studies of magnetic relaxation in nanoparticles of Ni dispersed in silica[END_REF]

χ' = χ o /[1+(ω m τ) 2 ], (1) χ
'' = χ o ω m τ/[1+(ω m τ) 2 ], (2)
where ω m = 2πf m , with f m = 1/τ m , the measuring frequency.

The characteristic relaxation time of each particle (τ), attributed to the thermally assisted reversal of the magnetic moments over the anisotropy energy barrier (KV) in the absence of an applied field and which is approximately valid for small applied field strengths is given by a Néel-Arrhenius law [START_REF] Goya | Magnetic dynamics of singledomain Ni nanoparticles[END_REF] (Néel-Brown expression [START_REF] Bedanta | Supermagnetism[END_REF]) as

τ = τ 0 exp(E B /k B T) = τ o exp(KV/ k B T). ( 3
)
Here V is the particle volume, T is the temperature, K is the effective uniaxial anisotropy, k B is the boltzmann constant and τ o is the inverse of the particle attempt frequency (f o = τ o -1 ) which represents the particle's attempt to jump between opposite parallel directions to the magnetisation's easy axis (for superparamagnetic systems typical values of τ o are ~10 -9 to ~10 -11 ) [START_REF] Goya | Magnetic dynamics of singledomain Ni nanoparticles[END_REF]. The blocking temperature (T B ) of the particles, below which the particle's moment will appear frozen on the time scale of the experiment can be isolated by a rearrangement of equation (3) (with T = T B ) as

T B = T a /ln(f 0 /f m ) (4)
where T a is equal to KV/k B and τ = τ m = 1/f m . It is known that as the effects of interparticle interactions become increasingly prevalent, then equation (4) will become modified into the form [START_REF] Singh | ac susceptibility studies of magnetic relaxation in nanoparticles of Ni dispersed in silica[END_REF] T B = T a /ln(f 0 /f m ) + T 0 (5) known as Vogel-Fulcher law, where T 0 quantifies the effects of the interparticle interactions. The aforementioned equations are based upon the assumption of uniform particle size. For distributions of particle sizes as in real samples, an average particle size is often assumed as in previous [START_REF] Singh | ac susceptibility studies of magnetic relaxation in nanoparticles of Ni dispersed in silica[END_REF][START_REF] Singh | Nickel-silica nanocomposite: Variation of the blocking temperature with magnetic field and measuring frequency[END_REF] as well as in the present work. For improved accuracy the equations may alternatively be modified appropriately to take account of particle size variation for improved accuracy as previously reported [START_REF] Gittleman | Superparamagnetism and relaxation effects in granular Ni-SiO_{2} and Ni-Al_{2}O_{3} films[END_REF]. By fitting the Néel-Arrhenius and the Vogel-Fulcher laws to the measured datasets, the effects of interparticle interactions may be analysed and quantified.

III. Results and Discussion

Ni(OH) 2 is an antiferromagnetic material with a bulk Néel temperature of ~26K [START_REF] Suzuki | Spin-glass phase and re-entrant spin-glass phase in diluted three-dimensional Ising antiferromagnet NicMg1-c(OH)(2)[END_REF]. While bulk antiferromagnetic materials have a roughly equal numbers of spin-up and spin-down magnetic moments leading to a compensation of the magnetic spins below the material's Néel temperature, on the nanoscale this may not always be true. Nanoscale antiferromagnetic particles may in fact acquire large magnetisations due to uncompensated magnetic spins, such as at the surface. The effects on the magnetic properties of the present samples owing to the presence of Ni(OH) 2 may hence be significant.

Figure 4 shows the hysteresis loops of the samples measured at 10K, 30K and 300K. It is seen that at 300K the samples appear to behave paramagnetically, displaying no appreciable coercivity or retentivity. As the temperatures of the samples are lowered to 30K and 10K from 300K, it is seen that the behaviours change from that of a paramagnetic signature to one of a mixed ferromagnetic/paramagnetic type of behaviour. In order to verify if the systems were behaving as ideal superparamagnets, the reduced magnetisations (M/M s ) were plotted as a function of (H/T). The classic law for superparamagnetism dictates that for a pure superparamagnetic sample, the resultant curves would have superimposed onto a universal curve [START_REF] Knobel | Superparamagnetism and other magnetic features in granular materials: A review on ideal and real systems[END_REF]. As evident from figure 5, the reduced magnetisation curves did not superimpose thus confirming the prevalence of a more complex underlying mechanism. Additionally, the failure of the samples to fully saturate with increasing field strength even at low temperatures is indicative of the mixed magnetic phases of the sample with the nonmagnetic phase (paramagnetic/antiferromagnetic) failing to achieve saturation even at 50 kOe applied field strength.

In testing for the presence of an exchange bias system, a number of loop shift measurements were conducted. At 300 K, the samples had a magnetic field of 10,000

Oe applied prior to the samples being cooled in the presence of the field down to 5 K.

Hysteresis loops were then measured from ± 10,000 Oe. The resulting curves are shown in Figure 6, in the insets of the figures an offset is discernable implying the presence of an exchange bias affect. To facilitate the delineation of the contribution from the different magnetic phases present, AC measurements were employed. To facilitate the further probing of the characteristics of the system, a series of AC susceptibility measurements were carried out at applied AC frequencies of 0.1 Hz, 2.5

Hz, 100 Hz, 500 Hz and 1000 Hz as shown in figure 7. Two peaks were clearly discernable in each of the AC susceptibility measurements at all frequencies. The perfect immobility of the lower temperature peaks with a changing measuring frequency discounts the presence of a blocking temperature at this point in the AC susceptibility measurements. The peaks may plausibly be attributed to the antiferromagnetic Néel temperature of the Ni(OH) 2 present in the system. In contrast, it is noted that the position of the higher temperature peaks in both the χ` and χ`` curves are observed to increase with increasing measurement frequency, a feature characteristic of a superparamagnetic blocking temperature. The peaks in the χ (χ` and χ``) versus temperature curves at higher temperatures may hence be attributed to the blocking temperatures of the pure Ni centres at the cores of the core-shell Ni-Ni(OH) 2 structures.

In addition to the AC measurements a series of Field-Cooled/Zero-Field-Cooled (FC/ZFC) measurements, with 1 Oe applied field strength were performed on both samples as shown in Figure 8. For both samples, the irreversibility point of the curves were observed at ~22K. A series of additional FC/ZFC measurements at further magnetic field strengths (10 Oe and 100 Oe) were also carried out (not all shown) and produced an similar irreversibility point at ~22K. It is noted that the slight upturn in the ZFC curve of sample 2 was not observed at higher field strengths and as such is likely attributed to the sample settling. The apparent invariance of the temperature of this point with varying applied field strengths appears contrary to what may be expected for a system containing superparamagnetic particles. In such a system it is known that the blocking temperature's position in the ZFC measurement will change with different applied field strengths. This effect has been attributed to the non-linear field of the unblocked particle's magnetisation and has been described elsewhere [START_REF] Suzuki | Spin-glass phase and re-entrant spin-glass phase in diluted three-dimensional Ising antiferromagnet NicMg1-c(OH)(2)[END_REF]. As no appreciable changes in the ZFC peak positions were observed with varying applied field strengths, this suggests the possible presence of a Néel temperature without the implication of a blocking temperature at ~22K.

However, with the blocking of the nanoparticle assemblages being evidenced in the AC susceptibility measurements, it is therefore likely that these affects are being masked by the Néel temperature peak in the static magnetic measurements. This observed phenomenon may be explained as follows, the blocking temperature can be defined using the following equation [START_REF] Knobel | Superparamagnetism and other magnetic features in granular materials: A review on ideal and real systems[END_REF].

ln τ m = ln τ 0 + B B T k KV 0 (6)
Here it can be seen that T B is dependent on the measurement time of the experimental apparatus and V 0 is the particle volume. T B is defined as the temperature at which the characteristic measuring time is equal to the intrinsic relaxation time associated with the nanoparticle's energy barrier. Typically, the measuring time is given a value of 100sec as this is a good approximation for the measurement time scale in a DC magnetometer. With an AC susceptibility measurement, the measuring time will be the inverse of the measurement frequency . In the AC measurements undertaken here, the values of τ m are therefore 10, 0.4, 0.01, 0.002 and 0.001 for the 0.1, 2.5, 100, 500 and 1000 Hz measurement frequencies respectively. The effect of this reduction in the measuring time will be to increase the system's measured blocking temperature from If a simple Néel temperature was being observed at ~22K then the reasons for an irreversibility point in the FC/ZFC curves may not be immediately apparent. It has however additionally been reported that in a mixed ferromagnetic/antiferromagnetic system an irreversibility point in the FC/ZFC curve may also be observed [START_REF] Acet | Coexisting ferro-and antiferromagnetism in Ni[sub 2]MnAl Heusler alloys[END_REF], Duman et al., 2002). The reasons for this are that for systems with both antiferromagnetic and ferromagnetic materials present, an irreversibility point in the FC/ZFC curves can occur at the lower transition temperature of the two materials present. Be that either the Néel temperature or the Curie temperature, in the present case it occurs at the Néel temperature. The only stipulation for this effect is that the applied field should be sufficiently small. This effect has been attributed to the preferential pinning of the spins into different configurations by the anisotropy of the antiferromagnetic phase. The FC/ZFC measurements can hence be used to determine if different forms of coexisting long range magnetic phases are present in the samples.

While the aforementioned effect may be present in the current samples under investigation, the FC/ZFC seperation in figure 8 is sufficiently exaggerated. It is therefore likely that a superposition of both effects (both a pinning at the material boundary and a blocking of the centres) brought about by the coexistence of Ni nanoparticles and a Ni(OH) 2 corona is being observed.

It is noted that the particle sizes studied here are below the single domain sizes of Ni (below which the particle becomes a single, uniformly magnetized magnetic domain, as the energy required to create and sustain domain walls is no longer favourable) which are reportedly in the region of ~55 nm [START_REF] Lu | Magnetic Nanoparticles: Synthesis, Protection, Functionalization, and Application[END_REF], ~43 nm [START_REF] Gong | Ultrafine particles of Fe, Co, and Ni ferromagnetic metals[END_REF]. Due to the proximity of the particles, the samples may experience a degree of interparticle interactions. The effect of the capping layer and hydroxide shell will likely be to lessen the likelihood of interactions somewhat by increasing the inter-core distances. In a real system there are many possible types of interactions which may be present such as exchange interactions, dipole-dipole, tunnelling exchange etc. as outlined by Bedanta et al [START_REF] Bedanta | Supermagnetism[END_REF].

The degree of interparticle interactions within the two samples is more fully analysed through the use of AC susceptibility measurements. Figure 7 shows the plots of χ` and χ``, against temperature. The higher temperature's peak positions, were observed to shift to higher temperatures with increased measuring frequencies, an effect also observed in other nanoparticle systems [START_REF] Shim | Effect of interparticle interaction on the magnetic relaxation in NiO nanorods[END_REF][START_REF] Singh | Nickel-silica nanocomposite: Variation of the blocking temperature with magnetic field and measuring frequency[END_REF][START_REF] Goya | Magnetic dynamics of singledomain Ni nanoparticles[END_REF]. The frequency dependence of the blocking temperature for isolated particles has been predicted by the superparamagnetic Néel model to follow the Néel-Arrhenius law.

This law (which is equivalent to equation 3) may be expressed in the form [START_REF] Goya | Magnetic dynamics of singledomain Ni nanoparticles[END_REF])

f = f o exp(-T a /T) ( 7
)
where f is the relaxation frequency (f=1/τ). This law may also be linearised into the equation of a straight line of the form ln(f) = (-T a )(1/T) + ln(f o ) where f is left to equal the measuring frequency f m , and T is left to equal the blocking temperature T B .

According to G.F.Goya et al, for both the in-phase and the out of phase components of χ, the thermally activated Néel-Arrhenius model is followed. The fitting of this equation to the measured datasets for χ'' is shown in figure 9. The steeper slope of the fitted curve for sample 2 implies that the energy barrier increases from sample 1 to sample 2, this is due to the fact that as the energy barrier E a is increased then so too does the blocking temperature. It is seen that T a = 747 K while f o is 4.11x10 11 Hz.

These values of T a and f o can be compared to previously reported values for other Ni nanoparticle systems, a T a value of 270 K and an f o of 2.6x10 9 Hz have been reported

for ~ 4nm Ni (~15% loading) nanoparticles also embedded in a nonmagnetic SiO 2 matrix [START_REF] Singh | Nickel-silica nanocomposite: Variation of the blocking temperature with magnetic field and measuring frequency[END_REF].

In examining the behaviour of sample 2, again the Néel-Arrhenius law is fitted to the measured dataset as shown in figure 9. For sample 2, the calculated values of T a and f o are 1449 K and 3.34x10 17 Hz respectively. As these values seem implausibly high for the system under investigation, the results for sample 2 are likely being affected by the presence of a degree of interparticle interactions which have been reported to increase the f 0 of the system. The fittings hence imply that for sample 2 the interactions are not simply a perturbation to the anisotropy energy barriers, but are in fact strong in relation to the barrier height.

The strength of the dipolar interactions between nanoparticles will fall relative to the cube of the distance between the centres of the particles. The interaction strength between two adjacent particles is approximately given by μ 2 /r 3 , with μ being 7 that the blocking temperatures for sample 2 are higher than those of sample 1 at similar frequencies. The reduced interactions from sample 1 may be related to the increased surface area of the particles which in turn may result in a thicker Ni(OH) 2 shell thus increasing the core interparticle distances.

To better quantify the effect of interparticle interactions the phenomenological Vogel-Fulcher law has been fitted to the experimental datasets of samples 1 and 2.

For magnetically interacting particles, a Vogel-Fulcher law is used as a modification of the Arrhenius law but this is only valid for the case where the blocking temperature >>T 0 . The law implies a characteristic ordering temperature T 0 which arises from the interactions within the system. If we consider the Vogel-Fulcher law and place T 0 = 0, then the Vogel-Fulcher law will simply reduce to the Arrhenius law implying a transition at T 0 = 0K. Previously, the Vogel-Fulcher law has also been successfully applied to spin glass type systems, where it can be used to account for the observed frequency dependence of the freezing temperature T f . For the present superparamagnetic system, T 0 is the temperature at which the relaxation time would diverge, thus signalling a transition, the Vogel-Fulcher law is consequently only valid across a restricted temperature range and breaks down in the proximity of T 0 . This law is reported to take the form T B =T 0 +T a /ln(f 0 /f m ) where T 0 measures the strength of the interparticle interactions. For the present systems under investigation 4.11x10 11 Hz is taken as the likely magnitude of f o . For sample 2, this model fitting as shown in figure 11 returns a value of 584 K for T a and 14 K for T 0. For sample 1, the value of T a is 750K (close to the 747K returned by the Néel-Arrhenius law) with T o being 0.17K (close to 0K), this shows the interparticle interactions to be negligible for sample 1 while remaining substantial for sample 2. The smaller value of T a for sample 2 is to be expected given the smaller particle sizes of sample 2 relative to sample 1, thus helping to confirm the validity of the results.

Additionally, the parameter Φ = ΔT B /T B Δlog 10 f m was calculated to examine the nature of the interparticle interactions. Here ΔT B is the shift in the blocking temperature peak as determined from the χ'' curves, over the Δlog 10 f m frequency interval [START_REF] Goya | Magnetic dynamics of singledomain Ni nanoparticles[END_REF][START_REF] Dormann | A DYNAMIC STUDY OF SMALL INTERACTING PARTICLES -SUPERPARAMAGNETIC MODEL AND SPIN-GLASS LAWS[END_REF]. In the present study the calculated value of Φ for sample 1 is 0.11 and is calculated to be 0.057 for sample 2. It has been reported that 0.05 < Φ < 0.13 for coupled nanoparticles with increasing values of Φ indicating a reduction in the coupling strength. For isolated nanoparticles, Φ = 0.13 while for spin glasses Φ is reported to be very small (0.005-0.05).

The larger strength of interactions in sample 2 relative to sample 1 seems to imply a thicker Ni(OH) 2 shell around the metallic core. The sizes of the Ni nanoparticle cores could be anything up to the sizes identified using TEM microscopy depending on the Ni(OH) 2 layer thickness. It has been reported that for a system with cubic anisotropy and K 1 <0, the value of K a which is the effective uniaxial anisotropy is related to K 1 through the relation K a =K 1 /12 [START_REF] Goya | Magnetic dynamics of singledomain Ni nanoparticles[END_REF]. If it is assumed that the value of K 1 for the system is actually the same as that of bulk Ni (-8x10 5 erg/cm 3 ) then using the equation (T a =K a V/k B ) and assuming spherical particles, we can back calculate the particle diameter for sample 1 to be 14.3nm. Since the TEM confirms the particle size to be ~32nm then this would suggest that approximately 9nm of the particle's radius is in fact OA and oxidised Ni in the form of Ni(OH) 2 . For sample 2 a particle diameter of 13nm is estimated using this approach implying an approximate 0.5nm inter-core distance. It is known that anisotropies well above that of bulk Ni can become manifest in magnetic nanoparticle systems [START_REF] Godsell | Magnetic properties of Ni nanoparticles on microporous silica spheres[END_REF] which would likely have the effect of reducing the particle diameter as calculated above. In spite of this, it still becomes apparent that the quantity of Ni(OH) 2 present is considerably more substantial in sample 1 than in sample 2 leading to stronger interactions in sample 2 as evidenced in the magnetic measurements. The increased surface area of sample 1 would likely have led to an increased rate of oxygen diffusion through the OA capping layer thus leading to an increased rate of sub-capping layer surface oxidation of sample 1 relative to sample 2. Sub capping layer hydroxide shells of Ni(OH) 2 fabricated using this method may hence provide a convenient and facile means of controlling interparticle interactions in similar nanoparticle assemblages for differing applications.

IV Conclusions

Ni/Ni(OH) 2 core/shell nanoparticles have been shown to represent a potentially interesting nanomaterial. This morphology was observed to facilitate varying degrees of interparticle interactions within the nanoparticle assemblage. The presence of hydroxides within the material also leads to a more complex magnetic system with a combined antiferromagnetic/ferromagnetic signature. Static (DC) and dynamic (AC) magnetic properties of the synthesised samples were compared and the degree of interactions within the individual nanoparticle assemblages were investigated. The phenomenological Néel-Arrhenius and Vogel-Fulcher laws were successfully fitted to the experimental datasets to analyse the interparticle interaction strengths. A clear transition temperature peak was visible for both of the samples in the AC susceptibility as well as DC FC/ZFC measurements and was attributed to the antiferromagnetic Ni(OH) 2 corona around the metallic particle cores. The degree of oxidation of the Ni cores was also linked to the nanoparticle surface area. This implies that more accurate control of the hydroxide layer thickness is achievable which in turn may facilitate a more precise control of the magnitude of interparticle interactions within similar nanocomposite assemblages.

  Figure 8.

  the average magnetic moment of the particles and r being the distance between the particles[START_REF] Telem-Shafir | Magnetization dynamics in arrays of strongly interacting magnetic nanocrystals[END_REF]. One means of analysing the degree of interparticle interactions lies in analysing the widths of the normalised χ`` curves. It has been reported that for an ensemble of non-interacting nanoparticles, the distribution of anisotropy energy barriers of the particles will typically determine the normalized width of the AC susceptibility versus temperature curves[START_REF] Telem-Shafir | Magnetization dynamics in arrays of strongly interacting magnetic nanocrystals[END_REF]. The decrease in the scaled curve widths of the χ'' curves of sample 2 relative to sample 1 are clearly visible in figure10for the 3 representative frequencies shown demonstrating that dipolar interactions are more prevalent in sample 2 than in sample 1. This seems to confirm the earlier conclusion from the Néel-Arrhenius model fitting to the sample 2 dataset. A further indication of the dipolar interaction strength lies in a comparison of the blocking temperatures of the samples, which are expected to increase as the interactions strength increases. It is seen in figure

Figure 1 :

 1 Figure 1: X-ray diffractograms of air dried Ni nanoparticles, peaks marked with

Figure 2 :

 2 Figure 2: TEM and HRTEM image of Sample 1 -34nm.

Figure 3 :

 3 Figure 3: TEM image of Sample 2 -bimodal 34/14nm particle size, . Figure 4: Hysteresis curves for samples 1 and 2 at 10K, 30K and 300K, inset shows zoomed in image about the origin for the 10K measurements. Figure 5: Reduced magnetisation curves (M/Ms) plotted as a function of Ms(H/T) for samples 1 and 2 inset shows zoomed in image about the origin for the measurements.Figure 6: Loop Shift measurements for samples 1 and 2, inset shows zoomed in

Figure 7 :

 7 Figure 7: Temperature variation of χ` and χ`` for samples 1 and 2.

Figure

  Figure 8: Field-Cooled and Zero-Field-Cooled curves for samples 1 and 2 with 1

Figure 9 :

 9 Figure 9: Néel-Arrhenius model fitting to the blocking temperatures obtained

Figure 10 :

 10 Figure 10: Scaled χ'' vs temperature for samples 1 and 2 at varying frequencies.

Figure

  Figure 11: Vogel-Fulcher model fitting to the blocking temperatures from the χ''

Figure 1 :

 1 Figure 1 : J. Godsell et al.

Figure 3 :

 3 Figure 3 : J. Godsell et al.

Figure

  Figure 4 : J. Godsell et al.

Acknowledgements

This work was supported by Science Foundation Ireland (SFI) under the Principal Investigator Programme Contract No. SFI-PI-06/IN.1/I98 at the Tyndall National Institute. Additional funding was provided from the EU FP7 Network of Excellence (NoE) project -Beyond CMOS Nanodevices for Adding Functionalities to CMOS -'NANOFUNCTION' (Grant No. 257375). Further support from SFI Grant No. 06/IN.1/I85 at the University of Limerick is additionally acknowledged. 412 413


E Brion

F Carlier

G Harel

V M Akulin

Non-holonomic quantum control

Keywords: numbers: 02.20.Sv, 02.30.Yy, 03.65.Fd, 03.67.Lx, 03.67.Pp Open-loop quantum control, coherence protection methods

whether they are published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Stimulated by the potential applications it may have in the futuree.g. in quantum chemistry, for controlling molecular dynamics and reactions, or in the very active field of quantum information processing [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF], and greatly inspired by the already well-established mathematical framework of classical control, an intense theoretical activity has been developing on quantum control within the last few decades [START_REF] Butkovskiy | Control of Quantum-Mechanical Processes and Systems[END_REF][START_REF]Introduction to Quantum Control and Dynamics[END_REF]. Two main types of control scenarios exist, commonly referred to as open-and closed-loop controls. The latter implies that the system to be controlled is frequently measured and further Confidential: not for distribution. Submitted to IOP Publishing for peer review 14 February 2011 processed in a way which depends on the measurement outcome [START_REF] Wiseman | [END_REF]5,6,7]. Openloop strategies merely consist in perturbing the system in a deterministic way without resorting to any information feedback.

In the context of open-loop quantum control, there exists a number of different tasks that have specific mathematical structures formulated in algebraic terms and require the development of specific algorithmic tools: the control of quantum evolution, of quantum state, or of density matrix. In the evolution control, the evolution operator of a closed system is designed through the application of a set of specifically tailored perturbations n α n (t) Vn . The most general controllability requirement For the system to be controllable, the operators Vn 's together with the natural Hamiltonian Ĥ0 must span the whole Lie algebra u (N ) and, when fulfilled, this requirement ensures that the evolution operator Û (t) is not bound to remain in the subgroup of natural evolutions exp Ĥ0 t i , t ∈ R as for the uncontrolled case, but can take any value in U (N ), depending on how the α n (t)'s are chosen. The system is then relieved from its dynamical constraints (In terms of classical mechanics, such a system is said to have no holomonic constraints).

One can identify two qualitatively different situations. In the case when the Hamiltonian Ĥ0 , the perturbations Vn , and the available set of the parameters α n are such that by choosing α n (t) one can perform any infinitesimal transformation within an infinitesimally short interval of time, that is when the control is locally exerted over the entire Lie group, one can force the quantum system to follow any predetermined evolution. In the other case, when the parameters α n are such that within an infinitesimally short interval of time dt one cannot perform all the infinitesimal transformations, and in particular the transformation I + i Ĥ0 dt inverse to the natural evolution suggested by Ĥ0 , the complete control is still possible in the case of a compact Lie group. However this control cannot force the system to follow any predetermined trajectory, but just can ensure an evolution which coincides with the predetermined one in a set of equally spaced time intervals, that is stroboscopically. In this paper, we concentrate exclusively on the last case.

Many theoretical methods, based on optimization, have been put forward for explicitly identifying an appropriate set of functions {α n (t)} associated to a given set of perturbations Vn and achieving a specific evolution target [8,9,10,11,12]. Most of these techniques rely on a known or intuitively guessed specific solution which can be further optimized with respect to an appropriate cost functional, through variational schemes: the straightforward exploration of the entire multidimensional space of the control parameters would indeed imply numerical work, the complexity of which grows exponentially with the dimensionality of the system. Other theoretical approaches [13,14] use decompositions of the target evolution to build an appropriate pulse sequence by taking advantage of the specific algebraic properties of the system. This kind of constructive technique is, however, restricted to very specific systems, which present strong symmetry properties.

In this paper, we review a control method, called the nonholonomic control [15,16,17], which, by contrast, allows one to build pulse sequences achieving any unitary evolution of a generic closed quantum system, which requires neither any prior knowledge of the sought solution, nore any specific symmetry property of the system. We consider the simplest physical scenario, restricting the physical resources to the minimum, i.e. only two physical perturbations Va , Vb which are alternately applied to the system during timings which play the role of adjustable control parameters. Given a specific evolution target, the appropriate control pulse sequence is derived via a twostep algorithm whose remarkable efficiency is due to interesting algebraic properties of random matrices and N th roots of the identity.

Two important features of this method are its universality -it can be applied to any quantum system which exhibits the required algebraic properties, and its feasibility -two generic perturbations of the system, such as the interaction with two classical fields, will indeed generally free the system from its dynamical constraints. One of the most topical applications of our control method is the implementation of quantum gates as specific evolution operators of physical systems considered as candidates for a quantum computer. A third important feature is that the approach can be done robust with respect to a large class of uncontrolled external perturbations.

The reliability of quantum control -or information processing in the context of quantum computation, is threatened by the so-called quantum errors which result from the interaction of the system considered with its environment. These errors surely represent one of the most difficult challenges to be taken up on the route towards quantum computation. Inspired by classical error-correction, Shor was the first to propose to use redundant encoding of information in the computer's Hilbert space to actively protect it against a given set of errors [18]. The classical idea is to repeat information to make it more likely to be correctly transmitted. Quantum-mechanically, one appends to the information-carrying system an auxiliary system, called ancilla, and transfer -or encode, the information in a subspace of the full Hilbert space of the total system. Such a subspace is called a quantum code when errors act on it in a detectable and correctable way: after some time, one then can diagnose whether an error occurred on the information stored in the quantum code and, if needed, correct it through a recovery procedure [19,[START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]. In the specific so-called non-degenerate case, errors map the code space onto orthogonal subspaces. This implies an inequality, called the quantum Hamming bound, on the number of correctable errors and the dimensionality of the total system. Note that adding ancilla introduces extra errors, which must also be taken into account. An important class of quantum codes are the stabilizer codes [20,21]. Another, very specific, kind of quantum codes are the so-called decoherence-free subspaces [22,23,24,25,26], which present the property of being naturally immune to errors: the mere existence of such decoherence-free subspaces is, of course, not guaranteed and actually assumes very specific symmetry properties from the errors affecting the system. A unified theoretical presentation of a great variety of such passive and active quantum error correcting approaches has been recently proposed in [27]. In the same spirit, topological protection methods take advantage of the high degree of symmetry of the system's Hamiltonian to safely encode information in its degenerate ground level [28,29]. Alternative to the quantum code approach, the decoupling pulse strategy [30,31,32,33,34,35,36,37,38,39,40,41] was inspired by composite pulses, initially developed in NMR [42,43,44]. This technique basically consists in frequently perturbing the system in such a way that errors do not have the time to corrupt information before being averaged out.

Theoretical proposals for quantum codes or decoupling pulses usually rely on rather strong assumptions on the system and the error model: in general, they are developed for ensembles of two-level systems, affected by independent errors. Though quite reasonable for, e.g., single-atom quantum computer scenarios in which one atom carries one qubit of information, this kind of assumptions is more questionable when qubits are actually distributed over ensembles [45,46]. Moreover, based on symmetry properties of SU (2), these methods fail to be translated to, e.g., systems of qutrits. It is therefore interesting and necessary to develop alternative universal error-correcting/preventing approaches, which allow to deal with arbitrary systems affected by errors which a priori do not present any symmetry. In this spirit, we review here two techniques which were built as extensions of the nonholonomic evolution control method. The first one is inspired from both the quantum code approach and the quantum Zeno effect [47]. It basically consists in protecting the information encoded in an arbitrarily chosen subspace of the system's Hilbert space through frequently repeating a cycle of specifically designed steps, i.e. coding, decoding and measurement-induced projection of the system's state onto the initial code subspace [48,49,50]. The second one is a decoupling method which enables us to reliably apply an arbitrarily prescribed unitary transformation on the system's state, despite the presence of a (slow) environment [51]. Both methods required the development of specific algorithms, which are presented here.

The paper is structured as follows. In Sec. 2 we present the nonholonomic control technique for closed quantum systems -the principles as well as the algorithm used for explicitly calculating the control timings, and discuss a few applications in quantum computation. In Sec. 3, we present two extensions to coherence protection against firstorder effects of the environment. For both techniques, the necessary algorithmic tools are explained and numerical examples are provided. Finally, we conclude in Sec. 4 and give possible perspectives of the nonholonomic control method.

Non-holonomic quantum control of closed systems

In this section we present the nonholonomic control method which enables one to perform an arbitrarily prescribed unitary transformation on a closed quantum system through alternately submitting it to two controlled perturbations. In the first subsection the principles of the technique are explained, as well as the algorithm for computing the interaction pulse timings. In the second subsection, we discuss possible applications of our method to quantum computation: in particular, we provide a numerical example involving an atomic qutrit, and suggest other uses of our technique in compound systems for quantum information processing.

The framework of the method

We consider a closed quantum system described by an N -dimensional Hilbert space, whose natural dynamics is governed by the unperturbed Hamiltonian Ĥ0 . This system is alternately submitted to two constant perturbations Va and Vb . The full Hamiltonian thus takes the following pulsed form

Ĥc (t) = Ĥ0 + α (t) Va + [1 -α (t)] Vb = α (t) Ĥa + [1 -α (t)] Ĥb
where Ĥa ≡ Ĥ0 + Va , Ĥb ≡ Ĥ0 + Vb and α is a piecewise constant function from R into {0, 1} (see figure 1). After a generic sequence of K pulses of respective durations {τ k=1,...,K }, the evolution operator takes the form

Ûc (τ 1 , . . . , τ K ) = exp τ K i ĤK × . . . × exp τ 2 i Ĥ2 × exp τ 1 i Ĥ1 (1)
where Ĥk = Ĥa for odd k's and Ĥk = Ĥb for even k's.

It is a known result from Lie group theory that, when the operators i Ĥa , i Ĥb span the Lie algebra u(N ) itself -the so-called Bracket Generation Condition (BGC), any unitary Ûd ∈ U (N ) can be obtained as a finite product of elements of the singleparameter semi-groups exp τ i Ĥa , τ ≥ 0 and exp τ i Ĥb , τ ≥ 0 [52,53,[START_REF]Introduction to Quantum Control and Dynamics[END_REF]. In other words, for any unitary evolution Ûd ∈ U (N ), there exists an integer K ≥ 1 and a K-dimensional time vector τ ∈ R * K + such that Ûc (τ 1 , . . . , τ K ) = Ûd . Note that the value of K, though uniformly bounded due to the compactness of U (N ), is not specified by the theorem. It is, however, reasonable to expect it to be of the same order as the number N 2 of real parameters required to completely characterize a unitary matrix N × N . Our goal is to be able, for any given Ûd , to exhibit a pulse sequence achieving the control objective Ûc (τ 1 , . . . , τ K ) = Ûd . The first step is to check whether the system is nonholonomic, i.e. controllable, that is whether BGC is satisfied by the two matrices i Ĥa , i Ĥb . A first way consists in iteratively calculating their commutators of increasing orders until they span u(N ) [START_REF]Introduction to Quantum Control and Dynamics[END_REF]. When N increases this direct computation becomes, however, intractable. In that case, one can instead verify the following sufficient condition, suggested by V. Ka: the system becomes controllable, when the representative matrix of Ĥb in the eigenbasis of Ĥa has no off-diagonal zeros.

Once the BGC is checked, it is legitimate to look for a time vector τ achieving an arbitrarily prescribed transformation Ûd ∈ U (N ) one wants to impose on the system. A straightforward way would consist in directly trying to minimize the function

F κ ( s) ≡ exp sκ i Ĥa,b × . . . × exp s 2 i Ĥb × exp s 1
i Ĥa -Ûd for increasing timevector lengths κ's, stopping the procedure when the global minimum 0 is reached and setting K = κ min and τ = s min . This approach is, however, very uneasy and unlikely to succeed since F κ ( s) exhibits many local minima. In the following, we present an alternative and systematic method consisting of two steps : (i) first, one determines a sequence of N 2 pulses which achieves the identity transformation, (ii) then, one iteratively approaches a finite root of the desired transformation, Û 1/n * d , the full transformation Ûd being simply obtained by concatenating n * times the same control sequence.

At the end of an N 2 -control-pulse sequence, the overall evolution induced by Ĥc (t) is

Ûc ( τ ) = exp τ N 2 i Ĥb × exp τ N 2 -1 i Ĥa × . . . × exp τ 1 i Ĥa .
Our goal is first to determine an N 2 -component time vector τ 0 which achieves the identity, i.e. Ûc ( τ 0 ) = I. To this end, we look for an N -component time vector T such that exp

T N i Ĥb ×exp T N -1 i Ĥa ×. . .×exp T 1 i Ĥa = e iϕ/N × M-1 I 1/N M(2)
where e iϕ/N is an unimportant overall phase factor, M is a N × N unitary matrix and

I 1/N ≡      1 0 • • • 0 0 e i 2π N • • • 0 . . . . . . . . . . . . 0 0 • • • e i(N -1) 2π N     
is an N th root of the identity with nondegenerate spectrum. To compute T , we take advantage of the following algebraic property: for a nondegenerate unitary matrix

U ∈ U (N ) of characteristic polynomial N j=0 a j X j , U N = e iϕ × I ⇔ N j=0 |a j | 2 takes its minimal value 2. ( 3
)
To find T we therefore only need to minimize the function F T = N j=0 a j T 2 to 2, with respect to T . Unexpectedly, minimizing F T turns out to be very easy. The reason for that has roots in the Random Matrix Theory : according to Dyson's law, the eigenvalues of random matrices tend to repel each other, and are thus very likely to be almost regularly distributed on the unit circle, as those of a nondegenerate N th root of the identity. In other words, in the space of N × N unitary matrices, matrices of the type e iϕ/N M-1 × I 1/N × M are very abundant: a generic product of exponentials as in ( 2) is therefore often very close to an N th root of the identity, which can easily be reached by small variations of the timings. Once the N -component vector T is found, we concatenate T N times, thus getting the N 2 -component vector τ 0 such that

Ûc ( τ 0 ) = exp T N i Ĥb × exp T N -1 i Ĥa × . . . × exp T 1 i Ĥa N = e iϕ/N × M-1 × I 1/N × M N = e iϕ × I.
Forgetting the physically unimportant phase factor e iϕ , we hence have constructed a control pulse sequence which achieves the identity transformation.

Let us now show how, from τ 0 , one can build a vector τ * which achieves a finite (n * ) th root of the desired transformation. We first put the target matrix Ûd under the form Ûd = exp 1 i Ĥd , and define Û = exp i Ĥd . Obviously, one has Û =0 = I and Û =1 = Ûd . For 0 < 1 1, one can use the linearized form for the target transformation Û 1 = I + 1 i Ĥd + o 1 Ĥd , very close to the identity. On the other hand, allowing for a slight variation of the time vector τ 0 → τ 1 = τ 0 + δ τ 0 , one can write Ûc ( τ 1 ) under the linearized form Ûc (

τ 1 ) = I + ∇ Ûc ( τ 0 ) • δ τ 0 + o δ τ 0 Ĥa / , δ τ 0 Ĥb / . If the N 2 matrices ∇ Ûc ( τ 0 ) • e k
, where e k denotes the unit vector along the k th component of the time vector τ , span the full vector space of N × N anti-Hermitian matrices, one can easily find the time vector increment δ τ 0 such that Û

1 Ûc ( τ 0 + δ τ 0 ), that is 1 i Ĥd = ∇ Ûc ( τ 0 )
• δ τ 0 , through standard techniques of linear algebra. Repeating the same operation from the new starting point τ 1 with the new target transformation

Û 2 = Û 1 + 2 -1 i × Û 1 × Ĥd + o 2 -1 Ĥd , where | 2 -1 | 1, one can compute the new time vector increment δ τ 1 such that 2 -1 i Ûc ( τ 1 ) × Ĥd = ∇ Ûc ( τ 1 ) • δ τ 1 , provided that the N 2 matrices ∇ Ûc ( τ 1 )
• e k span the full vector space of N × N anti-Hermitian matrices. One continues until the rank of the set ∇ Ûc ( τ

) • e k , 1 ≤ k ≤ N 2 becomes
less than N 2 and therefore the time increment cannot be calculated any longer. This happens, say, at step l: setting n * = 1/ l -one can choose all k 's as inverses of decreasing integers, one finally gets a time vector τ * ≡ τ l for which Ûc ( τ * ) = Û 1/n * d . One implements the desired transformation Ûd through concatenating the sequence of

N 2 pulses τ * 1 , τ * 2 , . . . , τ * N 2
n * times, thus achieving the result claimed by the general theorem, with

K = n * × N 2 .
To finish this subsection, we evoke another possible parametrization of the evolution operator, i.e. by the amplitudes of the applied perturbations. To be more explicit, we now consider N 2 pulses (k = 1, . . . , N 2 ) of equal durations τ , during which Va and Vb are alternately applied to the system multiplied by different and controllable amplitudes {ξ k=1,...,N 2 }. The overall evolution operator now takes the following form

Ûc ξ = exp τ i Ĥ0 + ξ N 2 Vb × . . . × exp τ i Ĥ0 + ξ 1 Va .
The same two-step procedure applies, as before. One first looks for the N -component amplitude vector Ξ such that

exp τ i Ĥ0 + Ξ N Vb × . . . × exp τ i Ĥ0 + Ξ 1 Va = I 1/N , through minimizing to 2 the function F Ξ = N j=0 a j Ξ 2 with respect to Ξ,
where N j=0 a j Ξ X j is the characteristic polynomial of the N -exponential product. Concatenating Ξ N times yields the N 2 -component amplitude vector ξ 0 which satisfies Ûc ξ = I up to a global phase factor. Secondly, one applies the same kind of successive approximations as in the previous paragraph, by iteratively varying the amplitude vector ξ 0 → ξ 1 = ξ 0 + δ ξ 0 → . . . in order to successively achieve n th roots of the desired transformation Ûd for decreasing n's: one stops at step l, when the rank of the set = Ûd .

∇ ξ Ûc ξ l • ε k , 1 ≤ k ≤ N 2 becomes less than N 2 ( ε k denotes the unit

Applications to Quantum Computation

One of the basic prerequisites to Quantum Computation -the so-called DiVincenzo's criteria [54], is the ability to universally process the information encoded in the state of the system chosen as a computer, i.e. to induce any prescribed unitary evolution of the system chosen as a computer. Our technique precisely enables the experimenter to reach that goal, provided that he or she is able to alternately impose two different physical perturbations which check the BGC.

As an example we consider here an atomic qutrit consisting of a hydrogen atom, whose Hilbert space is restricted to three degenerate hyperfine states (see figure 2)

|0 ≡ |n = 1; L = 0; S = 1/2; I = 1/2; F = 1, m F = 1 , |1 ≡ |n = 1; L = 0; S = 1/2; I = 1/2; F = 1, m F = 0 , |2 ≡ |n = 1; L = 0; S = 1/2; I = 1/2; F = 1, m F = -1 .
If a static magnetic field B 0 = (B 0,x , B 0,y , B 0,z ) is applied to the atom, the Zeeman Hamiltonian takes the form ŴZ = -geµ B ˆ S • B 0 where g e 2 and µ B -9.274084 × 10 -24 J • T -1 are the electronic gyromagnetic factor and Bohr magneton, respectively, and Ŝl = 2 Σl , for l = x, y, z, with

S x,y S z •0 = •F= 1, m F =1• 1• = F= 1, m F =0• •2• = •F= 1, m F =-1• S z D G S x,y •g = •F= 0 a) b) 0 20
Σx ≡ 1 √ 2    0 1 0 1 0 1 0 1 0    , Σy ≡ i √ 2    0 -1 0 1 0 -1 0 1 0    , Σz ≡    1 0 0 0 0 0 0 0 -1    .
Here, we implicitly assumed that the field is weak enough for the nonresonant couplings |0 , |1 , |2 ←→ |g = |n = 1; L = 0; S = 1/2; I = 1/2; F = 0 to be neglected. To be more explicit, we suppose that µ B B 0,x,y,z ∆E hf where ∆E hf 1.42GHz is the hyperfine splitting of the ground level of the hydrogen atom. Now, if one applies a harmonic time dependent magnetic field, oriented along the x-axis and whose frequency is slightly detuned by the amount ∆ from the transitions |0 ↔ |g and |2 ↔ |g , the effective resulting (second-order) Hamiltonian -obtained after adiabatically eliminating the essentially unpopulated ground state [55], couples the states |1 and |3 and can be put under the form Ŵ = (geµ B B)

2 8 ∆ × Γ, with Γ ≡    1 0 -1 0 0 0 -1 0 1   
where B denotes the amplitude of the magnetic field. Note that we do not take into account the electric field of the electromagnetic wave considered, since no electric couplings exist in the ground state hyperfine multiplicity. Furthermore, we again assume that B is weak enough, so that the nonresonance condition geµ B B 2 ∆ 1 is met. Finally, choosing two sets of values B × Γ which satisfy the BGC as can be numerically checked. As an example, we choose to achieve the target evolution

Ûd =    1 0 0 0 0 -i 0 -i 0    .
Our algorithm provides us with a sequence of 13 pulses represented of figure 2, for a total control period of 148µs. Note that this system has been previously considered in our work [51] to illustrate another method, i.e. our coherence protection method via decoupling. The numerical results reported here are, however, new.

The nonholonomic control technique can also be applied to larger systems, including compound systems: in [17], for instance, it was used to design a sequence of tailored electromagnetic pulses achieving a CNOT quantum gate in a system of two Cesium atoms in dipole-dipole interaction.

As the number of its elementary subsystems grows, the direct control of a compound system becomes, however, computationally harder and finally intractable: for instance, the control of an N -qubit quantum computer requires the control of ∼ 4 N physical parameters. To overcome this difficulty, one can, as suggested in [16], construct quantum devices from fully controlled elementary cells, of relatively low dimensionality, comprising a few information-carrying particles, typically three. The spatial arrangement of these cells as well as the way they connect to each other must be designed for the specific computational task the device has to perform. As an example, it was shown in [16] how to perform universal computation in a 9-qubit quantum processor made of three triads of Rydberg atoms as elementary cells.

Coherence protection and the nonholonomic control

Let us consider an N -level quantum system S, of Hilbert space H, interacting with its environment E plus an additional classical noise, and submitted to a time-dependent control Hamiltonian, designed by the experimenter. The total Hamiltonian of the system (S + E) can thus be put under the form

Ĥ (t) = Ĥc (t) + Ω (t) ⊗ I E + I S ⊗ ĤE + VSE
where Ĥc represents the time-dependent control Hamiltonian, Ω (t) is the classical noise -as, e.g., the interaction with a parasitic classical field or inaccuracies in the control Hamiltonian Ĥc , ĤE is the natural Hamiltonian of E and VSE is the interaction Hamiltonian between S and E. The operators Ω and VSE can be expanded on a basis of Hermitian traceless generators of su(N ) of norm one, denoted by Ĝm=1,...,N 2 -1 ,

Ω (t) ⊗ I E = 1≤m≤M ε m (t) Ĝm ⊗ I E VSE = 1≤m≤M ζ m Ĝm ⊗ Êm
where the Êm 's denote Hermitian traceless operators of norm one acting on the environment's Hilbert space, the ε m 's are real functions of time, and the ζ m 's are real numbers. Note that the number M of terms in the expansion checks M ≤ N 2 -1, since not all the generators Ĝm are always necessary in the decomposition. Note also that, as the Ĝm 's and the Êm 's have been assumed of norm one, the strength of the interactions is fully characterized by the ε m 's and the ζ m 's. We shall moreover assume that these quantities are bounded: there exists a finite real number Γ such that ∀m, |ε m | , |ζ m | Γ. In the interaction picture relative to the Hamiltonian Ĥc (t) ⊗ I E + I S ⊗ ĤE , the evolution operator of the system (S + E) satisfies

i ∂ t Ũ (t) = 1≤m≤M Gm (t) ⊗ ε m (t) I E + ζ m Ẽm (t) Ũ (t) Ũ (0) = I
where we introduced the (unit-normed) operators in the interaction picture Gm (t

) ≡ Û † c (t) × Ĝm × Ûc (t)
⊗ ε m (s)

I E + ζ m Ẽm (s) × ε p (s ) I E + ζ p Ẽp (s ) + o t 2 /τ 2 Z .
The reliability of storage/processing of information in the system S is threatened by the terms of order n ≥ 1 in the interaction picture, which represent a discrepancy from the controlled evolution Ûc (t).

At the end of this section, we discuss the relevance of correcting all orders of the perturbation series, and show that, indeed, in most cases of interest the first-order errorcorrection only needs to be considered. In the following two subsections, we focus on two strategies for suppressing the first-order term, which use ideas from the nonholonomic control technique. The first one is inspired by the quantum Zeno effect. It consists in stabilizing a subspace of the system's Hilbert space -which, therefore, becomes effectively decoherence-free, by the frequent repetition of a coding-decoding-incomplete measurement procedure. The coding-decoding steps are physically implemented via two symmetric pulse sequences involving only two different control Hamiltonians, as in the closed-system evolution control technique exposed in the previous section. Here, however, the full control of the evolution operator is not required. As a consequence, fewer control parameters, i.e. fewer pulses are needed: the coding and decoding sequences can thus be made quite short. The second strategy we present is more ambitious, since it aims not only at completely decoupling the system from the classical noise and the environment, but also at performing an arbitrarily prescribed evolution on its state. We show that this can only be achieved through a pulsed control Hamiltonian which takes at least three different values, one of which may be zero. We also suggest how to construct an appropriate sequence from the two-valued control pulse sequence one would get for achieving the same control objective for the system S considered closed, via the technique presented in the previous section.

The main feature of both methods is their universality: they can indeed be applied to any system exhibiting the required minimal algebraic properties and, in particular, not only to systems of qubits, as it is the case for most of the techniques proposed in the literature. Another important point about these methods is that they employ minimal physical resources in terms of control Hamiltonian: for the first one, we only need two controlled perturbations, three for the second one.

Multidimensional Zeno Effect and Coherence Protection

In this subsection, following the main idea of the quantum Zeno effect, we investigate how repeated measurements together with pulse sequences enable one to annul the firstorder term in (4). To be more explicit, we present a strategy which makes it possible to stabilize a subspace of the system's Hilbert space, therefore making it effectively decoherence-free. We want to emphasize that, here, we present our method in a more general context than in previous works. In [48,49,50], we indeed focused on unitary errors corresponding to, e.g., the action of parasitic classical fields. But, as we shall see in the following, the same method holds for the generic model described in the introduction of this section.

The principle of the method

The quantum Zeno effect takes place in a system which is frequently measured in its (necessarily known) initial state: if the time interval between two projective measurements is small enough, the evolution of the system is blocked with a very high probability, equal to 1 in the (unphysical) limit of continuous measurements. As we shall now see, the same kind of idea can be used to stabilize the information stored in a multidimensional subspace of a system's Hilbert space.

Let us first assume there exists an observable O of the system S, for which the I(≥ 2)-dimensional subspace C ⊂ H, of projector P , is an eigenspace. For simplicity, we shall assume that C ⊥ is also an eigenspace of O, of projector Q = I S -P (considering more than just two eigenspaces for the observable O would not make the demonstrations more difficult, but more cumbersome). Moreover, we suppose the total system (S + E) is initially in the state ρ (0) ⊗ ρe , such that P × ρ (0) = ρ (0) × P = ρ (0). If no control is imposed on S, i.e. Ĥc ≡ 0 which implies that Ĥ0 ≡ 0, the evolution operator (4) for (S + E) at time τ τ Z takes the form

Ũ (τ ) I + 1 i m Ĝm ⊗ τ 0 ε m (s) I E + ζ m Ẽm (s) ds - 1 2 m,p Ĝm Ĝp ⊗ τ 0 ds s 0 ds ε m (s) I E + ζ m Ẽm (s) × ε p (s ) I E + ζ p Ẽp (s ) + o τ 2 /τ 2 Z
and the state at time τ is Ũ (τ )× ρ (0)⊗ ρe × Ũ (τ ) † . Note that, for sake of consistency, we introduced the notation ρ (0) ⊗ ρe for the initial state in the interaction picture relative to Ĥc , which of course coincides with ρ (0) ⊗ ρe . Measuring the observable O yields the new (mixed) state

P ⊗ I E Ũ (τ ) P ⊗ I E (ρ (0) ⊗ ρe ) P ⊗ I E Ũ (τ ) † P ⊗ I E + Q ⊗ I E Ũ (τ ) Q ⊗ I E (ρ (0) ⊗ ρe ) Q ⊗ I E Ũ (τ ) † Q ⊗ I E whose first-order term in τ /τ Z is 1 i m P Ĝm P ⊗ τ 0 ε m (s) I E + ζ m Ẽm (s) ds, ρ (0) ⊗ ρe .
Assuming the Ĝm 's act orthogonally on the subspace C ⊂ H, i.e.

∀m ≤ M, P × Ĝm × P = 0,

washes out this first-order contribution. The state of (S + E) is therefore given by ρ (0) ⊗ ρe up to second-order terms in τ /τ Z . Repeating the same sequence of operations n times, and tracing out the environment to get the reduced state of the system S at time T = nτ , one obtains

ρ (T ) = ρ (0) + nτ 2 m,p
+ + o T 2 nτ 2 Z → ρ (0) ,
which means that the information initially stored in C ⊂ H is preserved through the frequently repeated measurement of O.

The situation we have just considered is, however, rather unlikely. It is indeed pretty strong to assume that the Ĝm 's act orthogonally on C and, at the same time, that C is an eigenspace of some obervable of the system. It is, however, sufficient to suppose that two distinct I-dimensional subspaces C, C exist, each of which presents one of these properties, i.e. C, is an eigenspace of an observable O, while C exhibits the orthogonality property. Since these two subspaces have the same dimension, they are isomorphic and there exists a unitary transformation such that C = ĈC. The orthogonality conditions (5) on the subspace C can be translated into an equivalent set of conditions on the coding matrix Ĉ , i.e.

∀m, P × Ĉ † × Ĝm × Ĉ × P = 0. (6)
We now apply a control Hamiltonian H c (t) which is non zero only for a very short amount of time τ c τ Z at the begining and the end of each period τ , i.e. just before and just after every single measurement. The time τ c is assumed so short that one can neglect the accumulated action due to the classical noise and the interaction with the environment during the control pulse sequences. We moreover assume that H c (t) is designed in such a way that it implements appropriate coding and decoding operations Ĉ and Ĉ † (we shall show later how to proceed technically). Taking the orthogonality conditions ( 6) into account, one can repeat the same calculations as above and finally get the reduced state of the system S at time T = nτ , that is after n cycles codingdecoding-measurement

ρ (T ) = ρ (0) + nτ 2 m,p χ 2 m,p Ĉ † Ĝm Ĉ P ρ (0) P Ĉ † Ĝp Ĉ -ξ 2 m,p P Ĉ † Ĝm Ĝp Ĉ P , ρ (0) + + o nτ 2 /τ 2 Z .
When n tends to infinity with fixed T (i.e. τ → 0), we again have ρ (nτ ) → ρ (0). Going back into the original picture, the initial state ρ (0) = P × ρ (0) × P has therefore been applied the overall controlled evolution U c (T c ) = C × C † n = I, while the first-order contribution of the noise and interaction with the environment has been washed away.

In other words, the information initially stored in C has been preserved.

To finish this paragraph, we consider the situation when S is a compound system comprising two subsystems, the information carrying subsystem I and the ancilla A, whose respective Hilbert spaces are H I of dimension I and H A of dimension A. Let us assume the ancilla is initially prepared in a state |α , which is a non-degenerate eigenstate of some observable O A of the ancilla. From what we have just shown above, if there exists an entangling unitary matrix Ĉ acting on H = H I ⊗ H A such that ∀m, Pα × Ĉ † × Ĝm × Ĉ × Pα = 0, where Pα ≡ Î ⊗ |α α| is the projector onto the Idimensional eigensubspace of O A C = H I ⊗ Span[|α ], the information initially stored in C can be protected with very high probability by the frequent repetition of the following procedure: (i) transfer of the information from C to C through the application of Ĉ, (ii) free evolution for a short period of time τ τ Z , (iii) decoding via the application of Ĉ † and finally (iv) projection onto C via the measurement of the ancilla in its initial state |α .

A lower bound on the dimension A of the ancilla's Hilbert space H A can be obtained by merely comparing the dimension of the Hilbert space H I ⊗ H A , i.e. dim H I ⊗ H A = I × A, with the dimension of the union Σ of the subspace C with all the subspaces Cm ≡ Ĝm C induced by the action of the different Ĝm 's on C ≡ ĈC, i.e. Σ ≡ C ∪ C1 ∪ C2 ∪ . . . ∪ CM . The largest possible dimension for Σ is obtained when all the Cm 's are orthogonal not only to C -they are by definition of the matrix Ĉ, but also to each other: then Σ ≡ C ⊕ C1 ⊕ . . . ⊕ CM and dim Σ = I × (M + 1). Finally, as

Σ ⊂ H I ⊗ H A , one has dim Σ ≤ dim H I ⊗ H A and A ≥ M + 1. ( 7
)
This condition, called the Hamming bound, clearly prevents us from correcting the full basis of the (N 2 -1) traceless Hermitian generators of su (N ), since one always has

M + 1 = N 2 = I 2 × A 2 > A.

Implementation of the coding matrix via nonholonomic control for low-dimension systems

The question now is: given a subspace C and a set of operators Ĝm=1,...,M , how can one compute and physically implement the "coding matrix" Ĉ satisfying (6) ? As a unitary matrix on H I ⊗ H A , Ĉ can, if it exists, i.e. when ( 7) is checked, be implemented via the nonholonomic control technique. Provided that two physical perturbations Va , Vb can be alternately applied to the system, which satisfy the BGC on the whole Hilbert space H I ⊗ H A -note that, here, Ĥ0 ≡ 0 so that Ĥa,b = Va,b , it is legitimate to look for the coding matrix under the form

Û t = exp t nc i Ĥnc × . . . × exp t 2 i Ĥ2 × exp t 1 i Ĥ1 ( 8
)
where Ĥk ≡ Va for odd k's and Ĥk ≡ Vb for even k's, and search the time vector t such that Û t meets (6). It is important to note that, here, the number n c of required parameters can be chosen of the same order as that of the conditions set by ( 6) on the terms of Û t , which is less than

N 2 = I 2 × A 2 .
The problem we face is actually much less demanding in terms of physical resources than the usual complete evolution control dealt with in the previous section, and a specific algorithm must therefore be developed to solve it.

The method we propose is actually a mixture of two algorithms. The first one allows us to explicitly compute an appropriate coding matrix Ĉ. We start from an arbitrarily chosen unitary matrix Ĉ0 which is then iteratively slightly varied, under the prescription that it remains unitary, Ĉ0 → Ĉ1 = Ĉ0 + δ Ĉ0 → Ĉ2 = Ĉ1 + δ Ĉ1 → . . . At each step, the increment δ Ĉ is calculated through minimizing the quantity

F ({δ m=1,...,M }) ≡ m Ĉ × P + δ m Ĝm × Ĉ × P 2
with respect to the δ m 's, and setting 6) are always better satisfied by the successive Ĉl , and, ultimately, completely checked by Ĉ ≡ Ĉ∞ . Our algorithm converges rapidly, and only a few steps are needed.

δ Ĉ = 1 2 M m=1 δ m Ĝm Ĉ. Conditions (
We are, however, less interested in the explicit expression of the coding matrix than in getting a physical way to implement it. We thus propose to mix the iterative algorithm we have just described above with the nonholonomic control approach to construct a time vector t such that Û t is an appropriate coding matrix. One first chooses an arbitrary time vector t 0 -the order of magnitude of the timings is given by the natural time scale of the perturbations. The matrix Û t 0 plays the role of Ĉ0 for the previous algorithm. Then, one slightly varies the time vector t 0 → t 1 = t 0 + δ t 0 , the time increment being determined by δ Ĉ0 = ∇ Û t 0 • δ t 0 . One repeats the same operation to determine δ t 1 such that δ Ĉ1 = ∇ Û t 1 • δ t 1 and set t 2 = t 1 + δ t 1 , then we compute δ t 2 from δ Ĉ2 = ∇ Û t 2 • δ t 2 , etc. In this way, one forces the unitary matrix Û t to follow the path Ĉ0 → Ĉ1 → Ĉ2 → . . ., while iteratively building up the desired time vector. Finally, Û t ∞ = Ĉ∞ = Ĉ. We shall not present more details about our search algorithm here, but refer to [48] for more information.

As an example, we consider as a toy model the protection of one out of five qubits, arranged in a random two-dimensional array, against all the M = 15 single-qubit errors σx ⊗ I ⊗ I ⊗ I ⊗ I,

σy ⊗ I ⊗ I ⊗ I ⊗ I, . . . , I ⊗ I ⊗ I ⊗ I ⊗ σy , I ⊗ I ⊗ I ⊗ I ⊗ σz .
as we are allowed by the dimensionality of the ancillary subspace A = 2 4 = 16, according to the Hamming bound. The model Hamiltonian we choose comprises an interaction part among the qubits Vint =

1 2 i =j k=x,y,z α k R 3 ij σ(i) k σ(j)
k and the (collective) interaction of the qubits with an external field k=x,y,z ζ k Σk , where Σk ≡ i σ(i) k . Here R ij denotes the distance between qubits i and j, the α k 's are constants characterizing the dipoledipole-like interactions between two qubits, and ζ k denotes the coupling of a qubit to the k-component of the exterior field. Running our algorithm for two different values of the total Hamiltonian Ĥa,b = Vint + k=x,y,z ζ

(a,b) k Σk , with α k R 3 ij ∼ ζ (a,b) k ∼ hν,
where ν is the typical frequency of the system, we find an 100-pulse sequence, reproduced in figure 3, achieving an appropriate coding matrix. The timings range from 4 × 10 3 × ν -1 to 1.6×10 5 ×ν -1 . Assuming that the whole control Hamiltonian can be reversed, including

(4) the interactions among qubits, the decoding matrix can be obtained by merely running the same sequence of pulses backwards with opposite Hamiltonians -H a,b . If this is not possible, the full control algorithm of the previous section must be run again, with the matrix Ĉ † as the target evolution. The example we have just addressed is completely new, but we also considered other applications and examples in previous works. For instance, we exhibited a coding matrix able to protect two qubits out of seven against 21 single-qubit errors plus 10 collective errors. This is indeed an important feature of our method that it does not rely on any specific symmetry of the errors, contrary to the independent error-model very often assumed in the literature about error-correction, but can protect information against any operator set Ĝm≤M provided ( 7) is satisfied. We also addressed more realistic specific physical situations, e.g. the protection of one qubit of information, stored in the spin of the external electron of a rubidium atom [48], its orbital part playing the role of the ancilla, as well as the protection of two qubits of information in a seven-spin molecule [49].

E a E b x y z V 12 V 13 V 14 V 15 0 0 8 10 6

Zeno coherence protection via random coding for large systems

The method presented above is adapted to low-dimensional systems. As the dimension increases, however, the algorithm we used to compute and implement the exact coding matrix becomes heavier and heavier, and is finally intractable for large dimensions. Fortunately, for high-dimensional compound systems, affected by physically reasonable errorinducing interactions, by which we mean the independent interaction of each subsystem with the environment plus binary interactions between subsystems, one can alternatively use random coding to check conditions (6) to a satisfactorily good acurracy, and therefore cancel the first-order effects in (4).

To be more specific, let us consider an n-qubit compound system comprising an i-qubit information-carrying system I (I = 2 i ), and an a-qubit ancilla A (A = 2 a ). We assume the error-inducing Hamiltonian takes the form

Ĥ (t) = α=x,y,z k=1,...,n σ(k) α ⊗ ε (k) α (t) ÎE + ς (k) α Ê(k) α + 1 2 α,β k =l χ k,l α,β σ(k) α σ(l) β ⊗ ÎE
comprising the 3n independent interactions of each qubit with a classical parasitic field and the environment, plus the 9n (n -1) /2 binary interactions among qubits. We therefore have M = 3n + 9n(n-1) 2 = 3n(3n-1) 2 ∼ 9n 2 2 different error operators Ĝm 's (i.e. the σ (k) α 's and the products σ

(k) α σ (l)
β ). Moreover, we assume that all the couplings have the same order of magnitude /T 2 , whence the Zeno range here is given by

τ Z = T 2 /M T 2 /n 2 .
Let us first note that the matrices Ĝm 's expressed in the natural (decoupled) basis are quite sparse. Typically, they contain 2 n nonzero terms, of order 1. Accordingly, in the same decoupled basis, the matrices P Ĝm P typically have 2 i terms of order 1. If we now apply a generic random unitary matrix Ĉ at the beginning of the free evolution period and its inverse Ĉ † at the end, we will smear out the nonzero terms over all 2 n states of the entire system. In other words, the matrices Ĉ † Ĝm Ĉ will now have ∼ 2 2n

terms of the same order ∼ 2 n /2 2n = 2 -n , while the matrices P Ĉ † Ĝm Ĉ P will have ∼ 2 2i nonzero terms of the same order ∼ 2 -n . When n → +∞ for a fixed i (i.e. when a → +∞), the conditions (6) are perfectly met. It is, however, not enough for the correction method to work. Indeed, when n → +∞, the number M of errors to be considered also goes to infinity which could possibly result in a non zero accumulated action of the total error Hamiltonian: fortunately, since M is only polynomial in n, the overall action indeed tends to zero when n → +∞. As in the low-dimensional case, we suggest to physically implement the random coding matrix Ĉ via the alternate application of two Hamiltonians Ĥa , Ĥb checking the BGC. But, here, there is no need for a precise design of the pulse sequence: a generic choice of a sequence of control timings t 1 , t 2 , . . . , t R of a reasonably short length R ∼ n will result in a completely generic random encoding transformation

Ĉ = e t R i ĤR × e t R-1 i ĤR-1 × . . . × e t 1 i Ĥ1
where, as usual, Ĥk = Ĥa for odd k's and Ĥk = Ĥb for even k's. The decoding matrix Ĉ † is simply obtained by running the same sequence of pulses backwards with the Hamiltonians -Ĥa , -Ĥb . It is important to underline that, though the precise choice of timings t k is not necessary to implement an appropriate coding matrix Ĉ, it is, however, critical for our method to be able to apply the exact inverse transformation.

As a first example, the method of random coding was applied to the protection of two qubits carried by two nuclear spins in a seven-atom molecule [50]. In the future, it should also allow to deal with much larger systems.

Universal dynamical decoupling from slow noise via the nonholonomic control

The method presented in the previous subsection allows one to protect the information stored in a system through adding a large enough ancilla and repeatedly applying a coding-decoding ancilla measurement cycle to the compound system. The coding and decoding steps are relatively short, compared to the nonholonomic pulse sequences required to solve the complete evolution control problem. The error-prevention is, however, based on a strong redundancy of the information to protect, due to the ancillaadding.

In this subsection, we aim at fully decoupling and controlling the evolution of an information-carrying system, without resorting to such redundancy, but only through nonholonomic pulse sequences.

3.2.1.

The general framework of the method Our objective is to design a control Hamiltonian Ĥc (t) such that, at the end of the control period [0, T c ], the first-order of (4) vanishes while the controlled evolution takes an arbitrarily prescribed value Ûd , a priori different from the identity, Ûc (T c ) = Ûd (9) ∀m,

t 0 Û † c (s) × Ĝm × Ûc (s) ⊗ ε m (s) I E + ζ m Ẽm (s) ds = 0.
Assuming ε m (s) and Ẽm (s) vary only very slightly on the control period, the second condition boils down to ∀m,

t 0 Û † c (s) × Ĝm × Ûc (s) ds = 0. (10)
(9,10) can be considered as the starting point of most of the noise-tolerant quantum control methods, i.e. dynamical decoupling and composite pulses. Our first goal is to determine the minimal resources required, in terms of Hamiltonian, to reach the double objective given by (9,10). The first naive attempt one could be tempted to make is to use for Ĥc (t) the same two-valued pulsed form as in Sec. II. A NoGo theorem [51] however claims that universal control and protection cannot always be simultaneously performed in the two-valued control Hamiltonian approach. As we shall now show, (9,10) indeed cannot be universally satisfied by a two-valued pulse sequence for all Ĝm 's and Ûd 's. To show this, we explicitly construct for any two-valued Hamiltonian a noise operator which cannot be universally eliminated. Before giving the details of the proof, let us introduce a few useful definitions and notations. We first define the operators Ĉn ≡ We introduce the superoperators Ĥk whose action on any operator X is given by H k X ≡ Ĥk , X . We can thus write e +ε Ĥk Xe -ε Ĥk = +∞ l=0

ε l l! H l k X. Since Ĥa , Ĥb -Ĥa n = Ĥb , Ĥb -Ĥa n = Ĥk , Ĥb -Ĥa n
for every k, we can moreover set Ĉn ≡ H k Ĥb -Ĥa n . We can now start the proof by evaluating the first-order effect of Ĉn , as noise source, according to (10):

t 0 Û † c (s) × Ĉn × Ûc (s) ds = K k=1 t k 0 Û † k-1 e is Ĥk H k Ĥb -Ĥa n e -is Ĥk Ûk-1 ds = K k=1 Û † k-1 t k 0 +∞ l=0 (is) l l! H l+1 k Ĥb -Ĥa n Ûk-1 ds = -i K k=1 Û † k-1 e it k Ĥk Ĥb -Ĥa n e -it k Ĥk -Ĥb -Ĥa n Ûk-1 = -i K k=1 Û † k Ĥb -Ĥa n Ûk + i K k=1 Û † k-1 Ĥb -Ĥa n Ûk-1 = -i Û † c (T c ) Ĥb -Ĥa n Ûc (T c ) + i Ĥb -Ĥa n .
The contribution we have just obtained does not depend on the specific timing parameters t k 's but only on the final operation Û (T c ) = Ûd . Moreover, when Ûd and Ĥb -Ĥa do not commute, this contribution is nonzero at least for n = 1 and possibly for other n > 1. Note that 1 ≤ rank Ĉn , n > 0 ≤ N -1. Hence, for any Hamiltonian successively taking the two values Ĥa and Ĥb , we have found a noise source Ĉ1 = 1 2 Ĥa + Ĥb , Ĥb -Ĥa = 0 which cannot be universally eliminated for all operations. This completes the proof: universal control and universal protection cannot always be simultaneously performed in the two-valued control Hamiltonian approach. Now the question is how to design a decoupling/control pulse sequence: in the following, we suggest a way to construct an appropriate three-valued pulse sequence, through simply adding "waiting periods" to a two-valued pulse scheme achieving the control objective (9) obtained via the technique exposed in Sec. 2. Suppose we have a sequence of timings {t 1 , t 2 , . . . , t K } which yields the desired evolution Ûd

Ûd = exp t K i Ĥb × . . . × exp t 2 i Ĥb × exp t 1 i Ĥa .
The first-order contribution Ĝm of the operator Ĝm over the control period can be put under the form Ĝm = K k=1

Û † k-1 ĝm,k Ûk-1 where ĝm,k ≡ t k 0 exp -s i Ĥk Ĝm exp s i Ĥk ds. If we now allow for waiting periods τ k=1,...,K between each pulse, during which the Hamiltonian is set to zero -this corresponds to a third value Ĥc = 0 of the control Hamiltonian, the overall controlled evolution is not changed, but the first-order contribution of the operator Ĝm over the control period 0, K k=1 (t k + τ k ) now takes the form δ Ĝm ({τ k }) ≡ Ĝm + K k=1 τ k × Û † k Ĝm Ûk which is a linear function of the waiting times. Assuming that, for all m = 1, . . . , M, each set of operators Û † k Ĝm Ûk , k = 1, . . . , K spans the entire space of N × N traceless Hermitian operators, then it is possible to find sets of parameters τ (m) k which, independently, annul the linear combinations δG m , that is δ Ĝm τ (m) k = 0. For the same set of parameters {τ k } to simultaneously annul all the quantities δ Ĝm ({τ k }) , m = 1, . . . , M, it seems reasonable to impose that K must be of the same order as M × N 2 , that is, for M = N 2 -1 corresponding to the protection against all possible error operators Ĝm , K must be of the order of N 4 .

Hence, to compute the appropriate control sequence, we suggest to apply the following two-step algorithm. First, given two Hamiltonians Ĥa and Ĥb which satisfy the BGC, we compute the L N 2 timings t * l=1,...,L which achieve Û 1 N 2 d according to the method described in Sec. II. The same sequence is concatenated N 2 times to yield the K = L × N 2 N 4 timings {t k=1,...,K } which yield Ûd . Then, we look for the waiting times {τ k } by direct minimization to zero of the functional

F ({τ k }) ≡ M m=1 δ Ĝm ({τ k }) 2
constructed as the sum of the norms of the actions of all possible errors accumulated during the control period. In practice, since the waiting times must be positive, K should be chosen slightly larger than N 4 and the use of linear programming methods or other minimization techniques is required. Of course, any such technique will only find an all-positive solution if one exists. If this is not the case, however, one can simply repeat the algorithm with different initial timings and try again.

Control and protection of an atomic qutrit

As an example, we applied our method to the same system as considered in Sec. II, i.e. an atomic qutrit consisting of three degenerate hyperfine states of a hydrogen atom's ground level [51]. The goal was to apply the same evolution operator as previously

Ûd =    1 0 0 0 0 -i 0 -i 0   
to the system while completely decoupling it from any arbitrary, though slow enough, classical noise and/or environment. Running our algorithm, we find a control sequence consisting in 208 pulses, whose durations range from 2.5µs to 690µs for a total period of control T c 6.58ms. Though the pulse timings and magnetic field intensities are experimentally feasible, the time duration required for the whole gate may seem quite long; it must, however, be recalled that we are working with the hydrogen atom ground state which is very stable indeed. If noises are too large -and therefore their accumulated action on the whole sequence too big, higher levels may also be considered, which could offer shorter gate timescales.

Higher-order correction

In this last subsection, we address the question of error-correction beyond the first order of the perturbation theory. On first thoughts, it seems desirable to try and correct all the orders of the perturbative expansion of the evolution operator. High-order correction, however, has a cost, that is the lengthening of the control pulse sequence, which, in turn, also increases the errors accumulated in the system. The necessary compromise one has to find between these two contradictory effects sets an upper limit on the order of correction one should consider to effectively reduce the impact of the errors to its minimum. In the following paragraphs, we propose a quantitative study of these different aspects.

To begin with, let us assume that the first-order error contribution in (4) has been suppressed thanks to the control Hamiltonian Ĥc (t) as described in the previous subsection: the second-order contribution is now the leading term in the expansion, hence the most dangerous, and must therefore be compensated for. Still assuming the classical noise and environment evolve slowly enough on the control period, we must therefore add (N 2 -1)

2 matrix constraints on Ĥc In the same way, assuming errors are suppressed up to the n th order, by a control sequence of length K ∼ N 2(n+1) , the remaining error contribution is of order (n + 1) and given by E n ∼ Tc τ Z n+1 × 1 (n+1)! where the control period can be estimated from the average duration t of one pulse, i.e. T c K × t ∼ N 2(n+1) × t, whence

E n ∼ λ n+1 × N 2(n+1) 2 (n + 1)!
where λ ≡ t τ Z corresponds to the typical action of the errors in units of , and can also be viewed as the error probability amplitude during one control step.

To be more specific, let us consider a system of d qubits, for which N = 2 d . Given λ and d, the remaining error E n reaches its minimum E min (λ, d) for a certain order n = n min (λ, d). Figure 4 shows E min (λ, d) and the corresponding order of errorcorrection n min (λ, d) as functions of λ and d. On this figure, one sees, in particular, that, for a system of more than three qubits, the second-order correction becomes valuable only for λ e -20 , i.e. t 10 -9 × τ Z . The control step duration cannot be less than 10 -9 s, corresponding to the limit of feasible commutation times for the current laser devices. Conversely, for the second-order to be meaningful, the Zeno range τ Z must be larger than 1s or, equivalently, the accumulated action over one second must be less than . This requirement is very strong: basically it boils down to demanding that the system is almost perfectly isolated! Our quantitative study therefore shows that, for systems of more than three qubits, second-and higher-order error-correction is not relevant, since it would rather strengthen than correct the error contribution.

Conclusion

In this paper, we reviewed an open-loop control method, called the nonholonomic control technique, which allows one to impose an arbitrarily prescribed unitary evolution on a closed system subject to only two controlled perturbations checking the BGC. The general algorithm we developed to explicitly compute the appropriate pulse sequence achieving a specific control objective in a given specific physical situation was presented together with numerical examples, demonstrating the universality of our approach.

We also presented two extensions of our method which enable one to protect information storage and processing from the unwanted effects of the interaction with the environment (quantum errors). These two techniques aim at cancelling the firstorder action of uncontrolled quantum and classical arbitrary noises by frequently and strongly perturbing the system either through measurement -quantum-Zeno-effect-like protection method, or by strong and short decoupling pulses. Again, the strength of our methods consists in their generality and universality: no specific assumption is required either on the structure and algebraic properties of the system considered or on the errors which affect the system, as was shown by the numerical examples we chose to provide.

The perspectives of our work of course concern the field of quantum information where our control techniques could help performing safe quantum gates in physical systems which are currently considered for quantum computing implementations: singleatom systems and atomic ensembles, quantum dots, rare-earth ionic crystals, trapped polar molecules, etc. We would also like to study whether/how they could help enhancing physical phenomena such as, e.g., the Rydberg blockade in atomic ensembles. Finally, we want to investigate how, combined with measurements, our method may enable one to impose any arbitrarily chosen SL (N ) transformation on an N -dimensional system.

Figure 1 .

 1 Figure 1. Two-valued pulsed control Hamiltonian H c (t).

  vector in the direction of the k th amplitude). Setting ξ * = ξ l and n * = n l , one finally gets Ûc ξ * n *

Figure 2 .

 2 Figure 2. Control of an atomic qutrit : a) the three states of the hydrogen atom ground level, b) a pulse sequence achieving the desired transformation Ûd (see text).

8 ∆

 8 = -0.203mT, B (a) 0,y = 0.101mT, B (a) 0,z = -0.051mT, B (a) = 1.757mT, ∆ (a) = 0.892GHz, and B (b) 0,x = -0.203mT, B (b) 0,y = 0.203mT, B (b) 0,z = 0.101mT, B (b) = 1.015mT, and ∆ (b) = -0.892GHz for the physical parameters, one gets two Hamiltonians Ĥa,b ≡ -geµ B ˆ S • B (a,b) 0 + (geµBB (a,b) ) 2 (a,b)

0 εds

 0 s) I E + ζ p Ẽp (s) ds × τ m (s) I E + ζ m Ẽm (Tr ρe ε m (s) I E + ζ m Ẽm (s) × ε p (s ) I E + ζ p Ẽp (s ) (2Γ) 2When n tends to infinity with fixed T (i.e. τ → 0), ρ (nτ ) ∼ ρ (

Figure 3 .

 3 Figure 3. Protection of one out of five qubits against single-qubit errors. The qubits are arranged on a surface, interact via a dipole-dipole-like Hamiltonian and are alternately submitted to two different control fields E (a,b) responsible for the perturbations Va,b . A typical control pulse sequence achieving an appropriate coding matrix is reproduced on the figure.

  c (t) Ĝm Ûc (t) Û † c (s) Ĝn Ûc (s) = 0 which are equivalent to N 2 × (N 2 -1)2 scalar equations. Together with the control objective and the first-order conditions (9,10), this imposes ∼ N 6 requirements on the control sequence.

Figure 4 .

 4 Figure 4. Minimum error (up) and the corresponding order of the perturbation (down) as functions of the typical accumulated error during the control step λ and the dimension of the Hilbert space of the system N = 2 d .

  1 2 Ĥa + Ĥb , Ĥb -Ĥa

	Û0 ≡ I.
				n
					and the operators
	Ûn≥1 ≡ exp	t n i	Ĥn × . . . × exp	t 2 i	Ĥb × exp	i t 1	Ĥa

Acknowledgments

The authors thank F. Prats for fruitful discussions. E. B. dedicates this work to the memory of René Brion.