Simple Manipulation of a Microwave Dressed-State Ion Qubit
S. C. Webster, S. Weidt, K. Lake, J. J. McLoughlin, W. K. Hensinger
SSimple Manipulation of a Microwave Dressed-State Ion Qubit
S. C. Webster, S. Weidt, K. Lake, J. J. McLoughlin, and W. K. Hensinger
Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, UK (Dated: October 17, 2018)Many schemes for implementing quantum information processing require that the atomic statesused have a non-zero magnetic moment, however such magnetically sensitive states of an atom arevulnerable to decoherence due to fluctuating magnetic fields. Dressing an atom with an externalfield is a powerful method of reducing such decoherence [N. Timoney et al. , Nature , 185], even ifthe states being dressed are strongly coupled to the environment. We introduce an experimentallysimpler method of manipulating such a dressed-state qubit, which allows the implementation ofgeneral rotations of the qubit, and demonstrate this method using a trapped ytterbium ion.
A key component of any quantum information proces-sor (QIP) is a long-lived qubit, isolated from the environ-ment to protect against decoherence [1]. For QIP devicesbased on trapped ions the dominant source of decoher-ence is usually dephasing due to random fluctuations ofthe magnetic field surrounding the ions unless a ‘clock’qubit, formed from a pair of levels whose energy separa-tion is unaffected by a change in magnetic field, is used.In many circumstances however, such clock qubits can-not be used, either because they do not exist in the ionin question, or they are not compatible with the pro-cess by which multiple-ion quantum gates are to be per-formed. The static magnetic field gradient gate proposedby Mintert et al. [2] is a promising gate technology fromthe point of scaling up an ion based quantum computer.A static magnetic field gradient is applied to the ions,which serves two purposes; it allows individual address-ability of ions by a global microwave field [3], and it allowsmicrowave fields to produce changes to the external de-grees of freedom of the ions, mediating gate operations.The different states making up the qubit however are re-quired to have different magnetic moments, meaning aclock qubit cannot be used. Gates using this static fieldgradient method have been implemented, however thegate fidelity was limited by field-induced decoherence [4].Although the intrinsic coherence time of a qubit maybe short, methods exist to increase it, such as dynam-ical decoupling by applying a series of π pulses to thequbit [5, 6]. Recently, Timoney et al. [7] proposed andimplemented a scheme where two states with oppositemagnetic moments are dressed by continuous microwavefields to form a state whose energy has no field depen-dence; this state is then combined with a state whichintrinsically has no field dependence to form an effectiveclock qubit. A method of manipulating the dressed-statequbit was also described, which allows rotation around aspecific axis in the x − y plane of the Bloch sphere only.Timoney et al. [7] reported a two orders of magnitudeimprovement of coherence time using this dressed-statequbit, which should allow high-fidelity multi-qubit gatesbased on static field gradients to be performed.We describe a different method of manipulating thedressed-state qubit which allows direct rotation of the (a) ⇥ (11) ⇥ ⇥ (12) d xdt + ⇥ x = 0 (13) d xdt + ⇥ x = fm cos( ⇥ t ) (14) d xdt + ⇥ x = fm cos( ⇥t ) (15) = ⇥ ⇥ (16)2 | i| i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) | pcol i = ( a | i + b (1 p ) / | i / p P dark | pcol i / a | i + b (1 p ) / | i ˆ X | pcol i / b (1 p ) / | i + a | i| uncol i / b (1 p ) / | i + a (1 p ) / | i | i| i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) | pcol i = ( a | i + b (1 p ) / | i / p P dark | pcol i / a | i + b (1 p ) / | i ˆ X | pcol i / b (1 p ) / | i + a | i| uncol i / b (1 p ) / | i + a (1 p ) / | i |⇧⌥ + |⌃⌥| ⌥ + | ⌥|⌃ , ⇧⌥ + |⇧ , ⌃⌥|⇤⇤⌥ + |⇤⌅⌥ + |⌅⇤⌥ + |⌅⌅⌥ ⌃/⇥ | h , ⇧ ⌥ + | v , ⌃ ⌥|⌃ , ⇧ ⌥ + |⇧ , ⌃ ⌥ g = r ⌥ h⇤ V cav µ (1) g = s cA ij ⇧ ⌃V cav (2) ⌅ = cT L (3) V ⇥ r (4) V ⇥ r (5) V ⇥ r (6) V = C r , C = e (4 ⌃⇤ ) (7) ⌥ (8) ⌥ + ⌥ str + ⇥ (9) |⇧ , ⇧⌥ + e . i |⌃ , ⌃⌥ (10)1 ⌦ + µ w ⌦ µ w | i| i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) | pcol i = ( a | i + b (1 p ) / | i / p P dark ⌦ + µ w ⌦ µ w | i| i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) | pcol i = ( a | i + b (1 p ) / | i / p P dark ⌦ + µ w ⌦ µ w | i| +1 i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) | pcol i = ( a | i + b (1 p ) / | i / p P dark ⌦ + µ w ⌦ µ w ! +B ! B | i| +1 i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) 1 ! ! = 1 ⌦ + µ w ⌦ µ w ! +B ! B | i| +1 i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) 1⌦ + µ w ⌦ µ w ! +B ! B | i| +1 i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) 1 µ s) P r obab ili t y i on i n F = (b)(c) Partial STIRAP Partial STIRAPDressed state ⌦ + µ w ⌦ µ w | i| i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) | pcol i = ( a | i + b (1 p ) / | i / p P dark ⌦ + µ w t hold t ST ! B | i| +1 i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) 1 ⌦ + µ w t hold t ST ! B | i| +1 i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) 1 ⌦ + µ w ⌦ µ w | i| i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) | pcol i = ( a | i + b (1 p ) / | i / p P dark ! = t o↵ t w t DS ! = t o↵ t w t DS ! = t o↵ t w t DS ⌦ + µ w ⌦ µ w | i| i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) | pcol i = ( a | i + b (1 p ) / | i / p P dark ⌦ + µ w ⌦ µ w | i| i| ... i| ... i ! B ! ! + |"i e . i⇡ | i = a | i + b | i| i = p ( | i i | i ) |h uncol | i|| i = p ( | i + | i ) | pcol i = ( a | i + b (1 p ) / | i / p P dark FIG. 1: (a) The S / ground state of the Yb + ion consist-ing of the F = 0 state | (cid:105) and three F = 1 states |− (cid:105) , | (cid:48) (cid:105) and | +1 (cid:105) whose degeneracy is lifted by an applied magneticfield. The hyperfine splitting ω / π is 12 . ω +B / π is 13 . ω +B − ω − B ) / π is −
30 kHz. Resonant microwavefields can be applied to manipulate or dress the ion, and aradiofrequency field can drive transitions between the F = 1levels. (b) Rabi oscillations between the first-order magneticfield insensitive | (cid:105) and | (cid:48) (cid:105) states, with a Rabi frequency of2 π ×
342 kHz. (c) Schematic of the dressed-state pulse se-quence. A partial STIRAP process transfers the ion from | +1 (cid:105) to | D (cid:105) . The microwave Rabi frequencies are held con-stant for a time t DS , during which time manipulation of thedressed-state qubit can take place using the rf field. Finallythe STIRAP process is completed, transferring any popula-tion in the | D (cid:105) state to |− (cid:105) . The STIRAP pulses are Gaus-sian, of width t w and offset t off . qubit state about any axis in the x − y plane. In addi-tion to allowing more general rotations, this method issimpler to implement experimentally, removing require-ments on the setting of the initial relative phases of thedriving fields that are needed in the original manipulationmethod [7]. We demonstrate an experimental implemen-tation of the dressed-state qubit, and our manipulationscheme, using a single Yb + ion. a r X i v : . [ qu a n t - ph ] M a r To describe our modification to the idea first proposedby Timoney et al. [7], we start first by summarising themethod they have proposed, in order to highlight thedifferences.The creation and manipulation of the dressed-statequbit is presented here as applied to the ground state of Yb + , although the method is applicable to other sys-tems. The hyperfine structure of the S / ground state of Yb + is shown in figure 1(a) and consists of an F = 0level | (cid:105) , and three levels with F = 1 ( |− (cid:105) , | (cid:48) (cid:105) & | +1 (cid:105) ,labeled corresponding to their m F value). The degen-eracy of these levels is lifted by a static magnetic field.There are two classes of transitions within the system,microwave transitions between F = 0 and the different F = 1 levels, and radiofrequency (rf) transitions betweenthe F = 1 levels.By applying continuous microwave excitation to dressthe ion, magnetically sensitive states can be stabilisedagainst disturbance caused by magnetic field fluctua-tions. The three atomic states | (cid:105) , |− (cid:105) and | +1 (cid:105) aredressed by two microwave fields of equal Rabi frequencyΩ µ w resonant with the | (cid:105)↔|− (cid:105) and | (cid:105)↔| +1 (cid:105) transi-tions, resulting in a Hamiltonian H µ w = ¯ h Ω µ w ( | +1 (cid:105)(cid:104) | + |− (cid:105)(cid:104) | + h . c . ), setting the two microwave phases set tozero (all Hamiltonians are presented in the interactionpicture and after making the rotating wave approxima-tion). The eigenstates of the coupled system are [7] | D (cid:105) = 1 √ | +1 (cid:105) − |− (cid:105) ) (1) | u (cid:105) = 12 | +1 (cid:105) + 12 |− (cid:105) + 1 √ | (cid:105) (2) | d (cid:105) = 12 | +1 (cid:105) + 12 |− (cid:105) − √ | (cid:105) (3)and the Hamiltonian can be re-written in this basis as H µ w = ¯ h Ω µ w √ | u (cid:105)(cid:104) u | − | d (cid:105)(cid:104) d | ) . (4)Without the dressing fields, fluctuations of the mag-netic field would cause the | D (cid:105) superposition to precess tothe state ( |− (cid:105) + | +1 (cid:105) ) / √ | u (cid:105) + | d (cid:105) ) / √
2, however thedressing microwaves lift the degeneracy of | D (cid:105) , | u (cid:105) and | d (cid:105) so only the part of the fluctuation spectrum aroundthis splitting frequency of Ω µ w / √ | D (cid:105) . The dressing fields thus protect theostensibly field sensitive state | D (cid:105) from field fluctuations.The remaining state | (cid:48) (cid:105) does not couple to this dressedsubsystem without additional interactions. If the states | (cid:48) (cid:105) and | D (cid:105) are used as qubit states then the qubit phasewill be unaffected by magnetic field fluctuations (besidesthose bridging the energy gap).Timoney et al. [7] described a method to manipulatethe dressed-state qubit as follows. To first order in theapplied magnetic field, the transition frequencies linking |− (cid:105) , | (cid:48) (cid:105) and | +1 (cid:105) , ω − B and ω + B , are equal, so a single rf field of Rabi frequency Ω rf and phase φ rf will coupleall three of the F = 1 states. The resultant Hamilto-nian H = H µ w + H rf where the rf terms to add to themicrowave terms (4) are: H rf = ¯ h Ω rf e iφ rf |− (cid:105)(cid:104) (cid:48) | + e − iφ rf | +1 (cid:105)(cid:104) (cid:48) | + h . c . ) (5)= ¯ h Ω rf φ rf ( | u (cid:105) + | d (cid:105) ) (cid:104) (cid:48) | − √ i sin φ rf | D (cid:105)(cid:104) (cid:48) | + h . c . )(6)after rewriting in the dressed-state basis.The states |− (cid:105) and | +1 (cid:105) were already linked togetherby the microwave fields, so adding a second linkage be-tween them by applying an rf field results in a loopedsystem. This loop means that the resulting form of theinteraction between states is drastically affected by thephase of the rf, the different paths around the loop inter-fering. This interference means the phase φ rf controls theRabi frequency at which a specific rotation in the Blochsphere occurs (and also the Rabi frequency at which | (cid:48) (cid:105) is off-resonantly coupled out of the qubit subspace, tothe states | u (cid:105) and | d (cid:105) ). Setting the phase φ rf to π/ H rf = ¯ h Ω rf √ i ( | (cid:48) (cid:105)(cid:104) D | − | D (cid:105)(cid:104) (cid:48) | ). This is a σ y coupling, rotating the qubit about the y axis of theBloch sphere. When implemented experimentally, caremust be taken whenever the rf or microwave frequenciesare changed that φ rf is correctly set to π/ x − y plane of the Bloch sphere about whichthe state rotates (a σ φ = cos φ σ x +sin φ σ y coupling fromhereon; σ x and σ y being specific cases of this more gen-eral coupling). This gives a flexibility in qubit rotationthat is not present in Timoney et al. ’s method [7].Timoney’s method [7] as presented assumes that ¯ hω +B ,the separation in energy between | (cid:48) (cid:105) and | +1 (cid:105) , is thesame as ¯ hω − B , the separation between |− (cid:105) and | (cid:48) (cid:105) . Fora sufficiently large magnetic field however, the 2nd or-der Zeeman shift lifts this degeneracy to producing asignificant difference between the transition frequencies δω = ω +B − ω − B . Unless Ω rf (cid:29) | δω | an additional rf field isneeded, so both transitions can be resonantly addressed.This doubling of the numbers of rf fields could poten-tially greatly complicate experiments, for instance in thecase where this gate is extended to perform multi-qubitgates [7]. A two-ion Mølmer-Sørensen gate [8] would po-tentially require 8 fields of different frequency due to thiseffect, rather than 4.Here we present a simpler method of performing singlequbit gate operations. It requires only one rf field anddoes not require the relative phases of the driving fieldsto be set to specific values. Arbitrary σ φ couplings areobtained with a simple change of the rf phase.This method takes advantage of the non-equal frequen-cies ω + B and ω − B of the two rf transitions due to the 2ndorder Zeeman shift. If we have a single rf field, resonantwith | (cid:48) (cid:105) ↔ | +1 (cid:105) , then with the condition that the Rabifrequency Ω rf (cid:28) | δω | we can ignore directly driven tran-sitions from | (cid:105) to |− (cid:105) as off-resonant, changing the rfpart of the Hamiltonian (5) to H rf = ¯ h Ω rf e − iφ rf | +1 (cid:105)(cid:104) (cid:48) | + h . c . ) (7)= ¯ h Ω (cid:48) rf e − iφ rf | D (cid:105)(cid:104) (cid:48) | + h . c . )+ ¯ h Ω (cid:48) rf √ e − iφ rf ( | u (cid:105) + | d (cid:105) ) (cid:104) (cid:48) | + h . c . ) (8)in the dressed-state basis where Ω (cid:48) rf = Ω rf / √ rf (cid:28) Ω µ w then transitions from | (cid:105) to | d (cid:105) and | u (cid:105) are suppressed by the energy gap, and we are left witha resonant interaction between | (cid:105) and | D (cid:105) , with a Rabifrequency Ω (cid:48) rf . The rf field however now no longer links |− (cid:105) to | +1 (cid:105) so there is no loop and φ rf can be freelychosen without causing interference effects. Changes to φ rf produce arbitrary σ φ couplings as occurs in a driventwo-level system. Thus, as long as Ω rf (cid:28) | δω | , Ω µ w , asingle rf field resonant with | (cid:48) (cid:105) ↔ | +1 (cid:105) is able to manip-ulate the dressed state qubit, and by changing the phaseof the rf perform arbitrary σ φ rotations. Detuning the rffield would allow general qubit rotations, about any axisin the Bloch sphere.We demonstrate this manipulation method of thedressed state qubit using a single Yb + ion confinedin a linear Paul trap [9]. Preparation of the ion’s initialstate and measurement of its final state are performed onthe bare ion without the presence of the dressing fields.We will briefly describe these steps and the method usedto switch between the bare and dressed states, beforedescribing the dressed-state ion manipulation.After Doppler cooling, 369 nm light resonant with S / , F = 1 → P / , F = 1 clears population out of |− (cid:105) , | (cid:48) (cid:105) and | +1 (cid:105) and prepares the ion in | (cid:105) . Any decay ofpopulation into D / is returned to the S / ↔ P / cycleusing light at 935 nm.The state of the ion is measured using a fluorescencetechnique. This technique allows us to distinguish be-tween the ion being in the F = 0 state | (cid:105) or one of the F = 1 states |− (cid:105) , | (cid:48) (cid:105) and | +1 (cid:105) when using 8 µ W of369 nm light focused to a beam waist of 20 µ m and reso-nant with the S / , F = 1 → P / , F = 0 cycling transi-tion (provided the ion is repumped from D / ). If the ionis in one of the F = 1 states photons will be scattered anddetected using a photo-multiplier tube. However, if theion is in the F = 0 state the light is ≈ p r obab ili t y i on i n F = FIG. 2: Decay of the | D (cid:105) state. The ion is held in the | D (cid:105) state for a variable length of time. After the 2nd partial STI-RAP is completed, a π pulse swaps the population in states |− (cid:105) and | (cid:105) before readout. The lifetime of the dressed-state | D (cid:105) is 550 ms, for microwave Rabi frequencies during t DS of2 π ×
16 kHz. The peak microwave Rabi frequency during theSTIRAP was 2 π ×
25 kHz, and the pulses were characterisedby t w = 450 µ s and t off = 356 µ s. technique is limited by off resonant excitation which cancause the ion to transition between the fluorescing F = 1states and the non-fluorescing F = 0 state (and vice-versa) [10]. With our setup we currently achieve a detec-tion fidelity of up to ≈ | (cid:105) and one of the F = 1states, a microwave signal is applied to the ion using amicrowave horn positioned 4 cm from the ion’s position.Figure 1(b) shows a typical set of Rabi oscillationswith a Rabi frequency of 2 π ×
342 kHz obtained by driv-ing the magnetic field insensitive | (cid:48) (cid:105) ↔ | (cid:105) transitionwith microwaves at 12.6 GHz. In all the experiments re-ported here, each repeat of the experimental sequencewas started at the same phase of the mains cycle. Thedecoherence can be quantified by measuring the coher-ence times of qubits based on these transitions by per-forming a series of Ramsey split pulse experiments. Forthe clock qubit | (cid:105) - | (cid:48) (cid:105) a coherence time ∼ | (cid:105) - | +1 (cid:105) qubit had a coherence time of ∼
40 ms, illustrating the way that magnetic field fluctua-tions dominate the dephasing of field sensitive states.To determine the two Zeeman splittings in the F=1manifold, the precise frequencies of the three microwavetransitions were measured, and the magnitude of themagnetic field at the ion determined to be 9.80(1) G.From these frequency measurements we get a frequencydifference in the two Zeeman splittings of -29(1) kHz. Forthe F = 1 states the 2nd order Zeeman shift δω = ω +B − ω − B = − (2 g J µ B B/ (2 I +1)) / h ω = − π × .
31 kHz / G [12], matching the measured value.In order to successfully dress the ion we require twomicrowave dressing fields. To generate the required mi-crowave frequencies, we begin with two different low fre- P r obab ili t y i on i n F =
100 100.5 101t (ms)
FIG. 3: Rabi oscillations within the dressed state. Basedon the short time behaviour, the Rabi frequency Ω (cid:48) rf = 2 π × . quency signals (0-30 MHz) which are then individuallyamplitude modulated before being combined and thenshifted into the microwave domain by mixing with a12.6 GHz source before being amplified and sent to themicrowave horn.An interrupted stimulated Raman adiabatic passage(STIRAP) process [13] is used to controllably dress andundress the ion. The ion is prepared in | +1 (cid:105) (using amicrowave π pulse from | (cid:105) ) and a STIRAP sequencestarted by adiabatically modulating the Rabi frequen-cies Ω + µ w and Ω − µ w of the two microwave fields, as thoughto transfer the ion to |− (cid:105) . At the point at whichΩ + µ w = Ω − µ w the ion is in | D (cid:105) , and the Rabi frequenciesare then held constant, ‘pausing’ the STIRAP process. Aschematic of this is shown in figure 1(c). Once the qubitis to be measured, the Rabi frequency modulation is re-sumed, finishing the STIRAP to transfer the populationin state | D (cid:105) to |− (cid:105) from where it is transferred to | (cid:105) bya microwave π pulse before the bare state measurement.The efficiency of the STIRAP process depends on theprocess being adiabatic. For a peak microwave Rabi fre-quency of 2 π ×
25 kHz, and using Gaussian amplitudeenvelopes, a maximum transfer efficiency of ∼
85% wasobtained for a t w in the range 250-500 µ s; higher peakRabi frequencies should improve this transfer efficiency.All data presented here has t w = 450 µ s and t off = 356 µ sA first measure of the robustness of the dressed statequbit is to measure the lifetime of the dressed state | D (cid:105) .This can be determined by measuring the population ofthe | D (cid:105) state as a function of the STIRAP pause time,as shown in figure 2. A fitted exponential to the datapoints gives a lifetime of | D (cid:105) of 550 ms, with microwaveRabi frequencies Ω µ w = 2 π ×
16 kHz dressing the ion. Byimproving our frequency generation set-up we expect toincrease this time to well beyond a second.To coherently manipulate our dressed qubit an rf fieldtuned on resonance with either | (cid:48) (cid:105) ↔ | +1 (cid:105) or | (cid:48) (cid:105) ↔ |− (cid:105) (cid:47) /2 pulse separation time (ms) P r obab ili t y i on i n F = FIG. 4: Ramsey fringe within the dressed state. Two π/ π/ x − y plane of the Bloch-sphere (a σ φ rotation). From the fringe period, a detuning δ rf = 2 π ×
160 Hz is inferred. is required. The rf field is generated by sending a 5 V amp signal into a 3 turn coil positioned ∼ (cid:48) rf = 2 π × . δ rf (cid:28) Ω (cid:48) rf the π/ π/ σ φ couplings are simply implemented witha change of the phase of the rf field. Using a similarextension to the coupling method as proposed by Tim-oney et al. [7], our method could be extended to drivemulti-ion entangling gates.This work is supported by the UK Engineering andPhysical Sciences Research Council (EP/E011136/1,EP/G007276/1), the European Commission’s SeventhFramework Programme (FP7/2007-2013) under grantagreement no. 270843 (iQIT), the Army Research Labo-ratory under Cooperative Agreement Number W911NF-12-2-0072 and the University of Sussex. The views andconclusions contained in this document are those of theauthors and should not be interpreted as representing theofficial policies, either expressed or implied, of the ArmyResearch Laboratory or the U.S. Government. The U.S.Government is authorized to reproduce and distributereprints for Government purposes notwithstanding anycopyright notation herein. [1] D. P. DiVincenzo, Fortschr. Phys. , 771 (2000).[2] F. Mintert and C. Wunderlich, Phys. Rev. Lett. ,257904 (2001).[3] M. Johanning, A. Braun, N. Timoney, V. Elman,W. Neuhauser, and C. Wunderlich, Phys. Rev. Lett. ,073004 (2009).[4] A. Khromova, C. Piltz, B. Scharfenberger, T. F. Gloger,M. Johanning, A. F. Var´on, and C. Wunderlich, Phys.Rev. Lett. , 220502 (2012). [5] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. ,2417 (1999).[6] G. S. Uhrig, Phys. Rev. Lett. , 100504 (2007).[7] N. Timoney, I. Baumgart, M. Johanning, A. F. Var´on,M. B. Plenio, A. Retzker, and C. Wunderlich, Nature , 185 (2011).[8] A. Sørensen and K. Mølmer, Phys. Rev. A , 022311(2000).[9] J. J. McLoughlin, A. H. Nizamani, J. D. Siverns, R. C.Sterling, M. D. Hughes, B. Lekitsch, B. Stein, S. Weidt,and W. K. Hensinger, Phys. Rev. A , 013406 (2011).[10] A. H. Myerson, D. J. Szwer, S. C. Webster, D. T. C.Allcock, M. J. Curtis, G. Imreh, J. A. Sherman, D. N.Stacey, A. M. Steane, and D. M. Lucas, Phys. Rev. Lett. , 200502 (2008).[11] M. Acton, K.-A. Brickman, P. Haljan, P. Lee, L. Deslau-riers, and C. Monroe, Quantum Inf. Comp. , 465 (2006).[12] G. K. Woodgate, Elementary Atomic Structure (OxfordUniversity Press, 1980), 2nd ed.[13] K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod.Phys.70