Micromotion-enabled improvement of quantum logic gates with trapped ions
Alejandro Bermudez, Philipp Schindler, Thomas Monz, Rainer Blatt, Markus Müller
MMicromotion-enabled improvement of quantum logic gates with trapped ions
Alejandro Bermudez,
1, 2, ∗ Philipp Schindler, Thomas Monz, Rainer Blatt,
3, 4 and Markus M¨uller Department of Physics, College of Science, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom Instituto de F´ısica Fundamental, IFF-CSIC, Madrid E-28006, Spain Institute for Experimental Physics, University of Innsbruck, 6020 Innsbruck, Austria Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
The micromotion of ion crystals confined in Paul traps is usually considered an inconvenient nuisance, and isthus typically minimised in high-precision experiments such as high-fidelity quantum gates for quantum infor-mation processing. In this work, we introduce a particular scheme where this behavior can be reversed, makingmicromotion beneficial for quantum information processing. We show that using laser-driven micromotion side-bands, it is possible to engineer state-dependent dipole forces with a reduced effect of off-resonant couplings tothe carrier transition. This allows one, in a certain parameter regime, to devise entangling gate schemes based ongeometric phase gates with both a higher speed and a lower error, which is attractive in light of current effortstowards fault-tolerant quantum information processing. We discuss the prospects of reaching the parametersrequired to observe this micromotion-enabled improvement in experiments with current and future trap designs.
CONTENTS I. Introduction
Intrinsic and excess micromotion
Entangling gates based on micromotionsidebands
Experimental considerations
Conclusions and Outlook
I. INTRODUCTION
The possibility of harnessing the distinctive behavior ofquantum-mechanical systems to process information in new ∗ Email to: [email protected] ways has raised the interest of researchers for more than threedecades now. This has given rise to the multi-disciplinary fieldof quantum information processing (QIP) [1], which could,for instance, have an impact on current cryptographic proto-cols [2], or revolutionise our approach to solve long-standingproblems in quantum many-body physics [3]. Motivated bysuch remarkable applications, QIP has now turned into a ma-ture field where experimentalists are using different technolo-gies [4] to face the challenge of building registers of ever-increasing sizes, while trying to preserve and manipulate theirquantum features for ever-longer periods of time.Among these so-called quantum technologies , crystals oftrapped and laser-cooled atomic ions [5–7] have played a lead-ing role in the progress of QIP. Pioneering proposals to build aquantum-information processor based on trapped ions [8], andsuccessfully implemented in the laboratory [9], have openedan active avenue of research with the ultimate goal of buildinga large-scale trapped-ion quantum computer [10]. As empha-sised early in the literature [11], success in such an enterprisewould require (i) a careful assessment of the possible imper-fections of the quantum processor, which lead to errors in thecomputation, and (ii) a thorough study of the unavoidable cou-pling to an external environment, which degrades the quantumcoherence responsible for the advantages of QIP. The formeryields errors that can accumulate quickly since the informa-tion is not stored in classical binary variables [11], whereasthe latter yields an exponential decrease of quantum coher-ence with the size of the register [12].Despite such a daunting perspective, the subsequent devel-opment of quantum error correction showed that these diffi-culties can be overcome if (i) one encodes the information re-dundantly in an enlarged quantum mechanical system insteadof using a bare quantum register, and (ii) errors are detectedand corrected during the storage and processing of encodedstates [13]. Increasing levels of protection against noise canbe achieved e.g. by concatenating elementary quantum errorcorrecting codes, or by storing logical states in global, topo-logical properties of larger quantum many-body systems [16].It has been shown that fault-tolerant
QIP is possible providedthat errors, either due to imperfections of gates or to environ-mental decoherence, occur below certain critical rates. The a r X i v : . [ qu a n t - ph ] N ov particular threshold values depend on the details of the imple-mentation, the noise model, and the chosen encoding. For cir-cuit noise models, a common estimate for concatenated codesis around 10 − [15], whereas topological codes typically of-fer higher error thresholds up to about 10 − [16]. Essentially,once below the threshold, quantum error correction allows forslowing down the occurrence of errors at the level of the log-ical qubits, such that longer computations can tolerate noiseon the physical qubits at a much higher rate. Trapped ionshave already demonstrated remarkable progress in experimen-tal demonstrations of quantum error correction [17–19].In order to meet such a threshold, one must optimise thehardware (i.e. quantum technology) and the software (i.e.schemes to manipulate the quantum information), which canbe understood as a built-in error suppression . At the softwarelevel, one can mitigate decoherence by encoding the informa-tion in a section of the Hilbert space that is more robust tothe typical environmental noise, as occurs for decoherence-free subspaces [20, 21], and for the so-called clock-statequbits [22]. Regarding imperfections of gates, pulsed [23, 24]and continuous [25, 26] dynamical decoupling have also beenimplemented in ion traps. Another possible source of er-ror arises in certain quantum technologies that exploit addi-tional auxiliary (quasi)particles to mediate an entangling gatebetween distant qubits, since quantum/classical fluctuationsaffecting these (quasi)particles can introduce errors in thecomputation. Such is the situation with trapped ions, wherephonons serve as a quantum bus to generate entanglement,and thermal fluctuations lead to significant errors when theion crystals are not laser cooled to the groundstate [8]. In thisrespect, the development of gate schemes that minimise suchthermal sensitivity has been of paramount importance to thefield. These schemes typically use a state-dependent dipoleforce in the resolved-sideband regime, which forces the ionsalong a closed trajectory in phase space depending on the stateof the qubits, either in the σ z [27, 28] or σ φ [29, 31] eigenstatebasis, where σ φ = − cos ( φ ) σ y − sin ( φ ) σ x . This effectivelyleads to state-dependent multi-qubit geometric phases that canbe exploited to generate entanglement, which underlies the re-markably low errors that have been achieved in experimentsso far [32–34], with infidelities reaching values below 10 − .An increase of gate speed yields another clear route forfurther error suppression, as the environmental decoherenceaffecting the qubits, or other external sources of noise af-fecting the phonon bus, would have a smaller impact dur-ing a shorter computation. Schemes for ultra-fast entanglinggates based on concatenated resonant state-dependent kickshave been studied in detail [35], which abandon the resolved-sideband regime to avoid the associated limitations on the gatespeed. These schemes give a clear advantage provided thathigh laser repetition rates [36], and small laser intensity fluc-tuations [37], can be achieved in the laboratory. Pulse split-ting techniques have been implemented in order to increasethe number of pulses incident on the ion [38], increasing thusthe repetition rate towards a regime where ultra-fast gates areexpected to have small errors [36]. To overcome the stringentconditions on the laser intensity stability [37], dynamical de-coupling approaches may have to be applied in order to min- imise the error of each resonant state-dependent kick [39].In order to avoid these technical difficulties, but still getan increase on gate speed with respect to previous realisa-tions [28, 31], schemes based on state-dependent σ z -forceswith an increased laser intensity have also been studied [40],which take into account the leading-order corrections as oneabandons the resolved-sideband regime. In this case, such cor-rections correspond to a time-dependent ac-Stark shift, whichis usually neglected in the resolved-sideband limit [28], butstarts contributing as one increases the laser power, and thusthe gate speed [40]. The particular form of the σ z -force allowsone to take into account this term easily, finding robust pulsesequences for faster quantum gates [40]. Unfortunately, thestate-dependent laser forces of this scheme (i) cannot be im-plemented with clock-state hyperfine qubits [41], and (ii) havesome limitations for optical qubits in comparison to the en-tangling gates generated by σ φ -forces [42]. It would be thusdesirable to consider schemes to speed up entangling gatesbased on σ φ -forces valid for both hyperfine and optical qubits.Unfortunately, the leading-order corrections to the resolved-sideband limit correspond to a time-dependent carrier drivingthat interferes with the σ φ -force (see our discussion in Sub-sec. III B below), and thus compromises the geometric char-acter of the gate and the achievable fidelities.In this work, we show that σ φ i σ φ j -gates with higher speedsand lower errors can be achieved by exploiting the micromo-tion of ion crystals, namely a periodic motion synchronouswith the oscillations of the quadrupole potential that confinesthe ions in a Paul trap. We consider two different types ofmicromotion: excess and intrinsic micromotion. Excess mi-cromotion can be described as a classical driven motion of theions that lie off the r.f. null, either due to imperfections ofthe trap or to crystal configurations with equilibrium positionswhere the r.f. field does not vanish. The role of this excessmicromotion on entangling-gate schemes has been consideredpreviously, showing that (i) purposely-induced excess micro-motion can be exploited to address different ions in a crystalvia differential Rabi frequencies of secular sidebands [43]; (ii) micromotion sidebands can be exploited to increase the gatespeed with respect to schemes based on secular sidebands, insituations where the excess micromotion cannot be perfectlycompensated [34]; and (iii) pulse sequences for entanglinggates based on either standard normal modes [44, 45] or soli-tonic vibrational excitations [46], can be designed even in thepresence of the excess micromotion. With the exception ofRef. [44], the role of another type of micromotion in schemesof entangling gates, namely the intrinsic micromotion, hasremained largely unexplored. Intrinsic micromotion corre-sponds to a quantum-mechanical driven motion synchronouswith the r.f. frequency which cannot be compensated. Be-ing quantum-mechanical, the intrinsic micromotion has a dif-ferent impact on the gate schemes. In contrast to Ref. [44],where pulsed gate schemes are used to make the performanceof the gate equal to the ideal case where no micromotion ispresent, we explore in this work the possibility of actively ex-ploiting the intrinsic micromotion in order improve the gateperformance, both in speed and fidelity, beyond the values ofthe schemes where no micromotion is considered.This article is organised as follows. In Sec. II, we introducethe formalism that allows us to describe excess and intrinsicmicromotion in generic ion crystals confined by Paul traps.This formalism is the starting point to develop in Sec. III ageneral theory of laser-ion interactions in the regime of re-solved sidebands in presence of both excess and intrinsic mi-cromotion. The expressions obtained are then used to de-scribe the main differences of the schemes that generate state-dependent dipole forces using bi-chromatic laser beams, ei-ther tuned to the secular or to the micromotion sidebands. Wealso describe how these forces can be used to implement en-tangling gates, and discuss the speed and fidelity limitations ofvarious gate schemes, identifying a parameter regime where agate improvement can be obtained by exploiting the intrinsicmicromotion. In Sec. IV, we discuss the possible experimen-tal challenges in reaching such parameter regime. Finally, wepresent our conclusions and outlook in Sec. V. II. INTRINSIC AND EXCESS MICROMOTION
In this section, we start by reviewing the classical treatmentof micromotion for a single trapped ion in II A. This will al-low us to set the notation, and to explicitly define the notionsof intrinsic and excess micromotion in ion traps. Additionally,it will provide some results that will be useful in the subse-quent quantum-mechanical treatment in II B. The micromo-tion of a trapped-ion crystal is described in II C, which shallbe the starting point for the scheme of micromotion-enabledimprovement of quantum gates in the following section.
A. Classical treatment of micromotion for a single trapped ion
For the ease of exposition, we focus in this section on theelectric potential configuration and micromotion effects of anion confined in a standard linear Paul trap [6]. We note that asimilar analysis would apply to segmented linear traps [47],or to surface ion traps [48], which form a key central ele-ment in various scalable architectures for QIP under devel-opment [10]. At the end of this section, we will comment onthe analogies and possible differences for the micromotion inthese other traps.We consider an ion of mass M and charge Q , inside a stan-dard linear Paul trap formed by (i) a pair of end-caps sepa-rated by a distance 2 z along the trap axis (i.e. z axis), andconnected to d.c. potentials U ; (ii) four electrodes separatedfrom the axis by a distance r , and parallel to it, which are con-nected in pairs to either a d.c. potential V , or an a.c. potential V cos Ω rf t , where Ω rf is a fast r.f. frequency. Accordingly, theion is subjected to an oscillating quadrupole potential V q = κ U z (cid:18) z − ( x + y ) (cid:19) + V cos ( Ω rf t ) (cid:18) + r (cid:0) x − y (cid:1)(cid:19) , (1)where κ is a geometric factor that depends on the details ofthe electrodes. Here, we have assumed that the ions positionsfulfill | rrr | (cid:28) r , z , such that they lie close to the trap axis and trap center. In this way, we are neglecting corrections to thequadrupole potential, such as as small component of the alter-nating r.f. field along the direction of the trap axis.In addition to the ideal quadrupole potential (1), there canbe spurious potentials stemming from (a) potential varia-tions due to patch effects, or to unevenly coated (charged)electrodes with elements (electrons) coming from the oven(ionization process), and (b) asymmetries in the electrodeimpedances [49]. The former leads to spurious d.c. fields E dc that displace the ions from the nodal line of the a.c. potential,whereas the latter induce small phase differences in the a.c.electrodes ϕ ac , which give rise to an additional a.c. field. Thisfield can be approximated by that of a pair of parallel platesconnected to potentials ± V ϕ ac sin ( Ω rf t ) , and separated by2 r / ˜ α with ˜ α being another geometric factor that depends onthe trap configuration. These spurious effects thus lead to anadditional potential V s = − (cid:18) E x dc + V ϕ ac ˜ α r sin ( Ω rf t ) (cid:19) x − E y dc y − E z dc z . (2)The classical equations of motion for the ion correspond toa set of inhomogeneous Mathieu equationsd r α d τ + (cid:0) a α − q α cos 2 τ (cid:1) r α = f α ( τ ) , α ∈ { x , y , z } , (3)where we have introduced the dimensionless time τ = Ω rf t ,and the following dimensionless parameters a x = a y = − a z = − Q κ U Mz Ω , q x = − q y = − QV Mr Ω , q z = . (4)In addition, we get force terms in Eq. (3) due to the spuriouspotential (2), namely f α ( τ ) = QE α dc M Ω + δ α , x QV ϕ ac ˜ α Mr Ω sin ( τ ) , (5)where we have used the Kronecker delta δ α , β in front of thecontribution that stems from the electron-impedance asym-metries, which leads to the small phase difference betweenthe electrodes along the x -axis. The solution of these differ-ential equations builds on the solution r h α ( τ ) to the homoge-neous Mathieu equation (i.e. f α ( τ ) =
0) [50] by applying themethod of variation of constants [51]. Due to the periodicityof the equation, the solution can be expressed using the Flo-quet theorem as follows r h α ( τ ) = ∑ n ∈ Z C α n ( A α e i ( β α + n ) τ + B α e − i ( β α + n ) τ ) , (6)where A α , B α are constants that depend on the initial condi-tions, β α are the so-called characteristic exponents, and C α n are the Floquet coefficients. By substitution, one finds thatthese coefficients fulfill a recursion relation C α n + − D α n C α n + C α n − = , D α n = a α − ( β α + n ) q α . (7)In typical experimental realizations, the parameters (4) fulfill a α , q α (cid:28) , (8)and this allows one to solve the above recursion to the desiredorder of accuracy. To the lowest-possible order, one finds β α = (cid:113) a α + q α , C α ± (cid:96) = ( − ) (cid:96) q (cid:96) α C α (cid:96) (( (cid:96) − ) ! ) , (9)where we have introduced a positive integer (cid:96) to label the dif-ferent harmonics. Note that C α − (cid:96) (cid:54) = C α (cid:96) for general param-eters. However, this difference can be neglected to leadingorder in the small parameters (8). Imposing that r α ( ) = r α , d r α ( τ ) / d τ | τ = =
0, and C α =
1, we find that the homo-geneous solution describing the motion of an ion inside anideal Paul trap is r h α ( t ) = r α ξ α cos ( ω α t ) (cid:32) + ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α (cid:96) (( (cid:96) − ) ! ) cos ( (cid:96) Ω rf t ) (cid:33) , (10)where we have introduced the so-called secular frequencies ω α = Ω rf β α , (11)which are much smaller than the trap r.f. frequency ω α (cid:28) Ω rf .We have also introduced the parameter ξ α = + ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α (cid:96) (( (cid:96) − ) ! ) . (12)We can rewrite Eq. (10) as r h α ( t ) = r sec α ( t ) + r in α ( t ) , such thatthe ion in an ideal Paul trap displays slow oscillations at thesecular frequency described by r sec α ( t ) = r α ξ α cos ( ω α t ) , (13)accompanied by smaller and faster oscillations synchronouswith the a.c. potential r in α ( t ) = r sec α ( t ) (cid:32) ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α (cid:96) (( (cid:96) − ) ! ) cos ( (cid:96) Ω rf t ) (cid:33) . (14)These smaller oscillations are referred to as micromotion , andoccur roughly at multiples of the r.f. frequency (cid:96) Ω rf (i.e. mi-cromotion sidebands). To distinguish this behavior from theone stemming from the spurious potential (2), these fast oscil-lations r in α ( t ) are sometimes referred to as intrinsic micromo-tion [52], to highlight the fact that such a motion is intrinsicto the oscillating quadrupole of an Paul trap, even for an idealtrap design without any imperfection (2).The solution to the forced Mathieu equation (3) can befound using the method of variation of constants with the twoindependent solutions associated to Eq (6), namely r h α ( t ) = A α r α , ( t ) + B α r α , ( t ) . The complete solution is r α ( t ) = r sec α ( t ) + r in α ( t ) + r ex α ( t ) , (15) where the additional part due to the spurious potential is r ex α ( t ) = (cid:90) d τ (cid:48) ( r α , ( τ ) r α , ( τ (cid:48) ) − r α , ( τ ) r α , ( τ (cid:48) )) f α ( τ (cid:48) ) W α ( τ (cid:48) ) , (16)and W α ( τ (cid:48) ) = r α , ( τ (cid:48) ) d r α , ( τ (cid:48) ) / d τ (cid:48) − r α , ( τ (cid:48) ) d r α , ( τ (cid:48) ) / d τ (cid:48) is the Wronskian of the two solutions, which can be shownto be constant in this case W ( τ (cid:48) ) = − β α . When performingthe integrals, we keep only the slowly-varying terms, whichgive rise to the leading-order solution r ex α ( t ) = r driv α ( t ) (cid:32) + ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α (cid:96) (( (cid:96) − ) ! ) cos ( (cid:96) Ω rf t ) (cid:33) , (17)where we have introduced the following driven amplitude r driv α ( t ) = QE α dc M ω α + δ α , x q x r ϕ ac ˜ α ( Ω rf t ) . (18)We thus observe that the spurious potential (2) induces adriven motion (17) that is also synchronous with the r.f. fre-quency, and is thus another type of micromotion. Since it isnot linked to the secular motion, and can only be reduced bycompensating the spurious potential terms (2), this motion isusually referred to as excess micromotion [49]. Along thistext, we will use the wording micromotion compensation torefer to the compensation of the stray fields that produce ex-cess micromotion.As a consistency check, we note that to linear order in q α ,the complete solution (15) built from Eqs. (13), (14) and (17)coincides with the solution presented in [49], which includesthe secular motion and the first micromotion sideband. Thehigher-order powers of q α allow us to account for all highermicromotion sidebands. Note also that the intrinsic (14) andexcess (17) micromotion only occur in those trap axes where q α (cid:54) =
0. According to Eq. (4), micromotion in an ideal linearPaul trap only occurs in the transverse directions, as there is nor.f. field along the axial direction, such that q z =
0. However,for realistic experimental conditions that depart from this idealcase, there might also be axial micromotion q z (cid:54) =
0, as occursfor instance in segmented linear traps [52]. Accordingly, wewill consider the most general case, and allow for micromo-tion in all possible directions q α (cid:54) = , ∀ α ∈ { x , y , z } . The par-ticular microscopic expression of these parameters will gener-ally differ from Eq. (4), and depend on specific details of thetrap. For the excess micromotion (17), the driven amplitude r driv α ( t ) will differ from the ideal case (18), and also depend onspecific details of the trap. However, one can treat the micro-motion generically using Eqs. (13), (14) and (17), with genericparameters r driv α ( t ) , a α , q α only restricted to fulfill Eq. (8). B. Quantum-mechanical treatment of micromotion for asingle trapped ion
Since the ultimate goal of this work is to exploit the micro-motion to improve phonon-mediated quantum logic gates be-tween distant trapped-ion qubits, a full quantum-mechanicaltreatment of the secular vibrations and the intrinsic/excessmicromotion in a trapped-ion crystal will be required. Adetailed quantum-mechanical treatment of the secular vibra-tions and intrinsic micromotion for a single trapped ion hasbeen described in [6] using a formalism based on quantum-mechanical constants of motion [53]. We now use this for-malism to generalize the description to situations where ex-cess micromotion of a single trapped ion is also present.The quantum-mechanical Hamiltonian of the ion inside thePaul trap can be described as H = ∑ α (cid:18) M ˆ p α + K α ( t ) ˆ r α − MF α ( t ) ˆ r α (cid:19) , (19)where we have promoted the position and momentum toquantum-mechanical operators fulfilling [ ˆ r α , ˆ p β ] = i δ α , β .Here, we have introduced a time-dependent spring constant K α ( t ) = M Ω (cid:0) a α − q α cos Ω rf t (cid:1) , (20)and used the time-dependent forces (5) caused by the devia-tions (2) from an ideal Paul trap, transformed back into realtime F α ( t ) = Ω f α (cid:0) Ω rf t (cid:1) . (21)The Heisenberg equations of motion for this Hamiltonian leadto a quantum-mechanical version of the classical forced Math-ieu equations (3) for the position operator, namelyd ˆ r α ( t ) d t + K α ( t ) M ˆ r α ( t ) = F α ( t ) , α ∈ { x , y , z } . (22)We now construct an operator constant of motion by com-bining the position operator ˆ r α ( t ) fulfilling Eq. (22), with amode function u α ( t ) that evolves according to the solution ofthe homogeneous classical Mathieu equation (6), but with ini-tial conditions u α ( ) = , d u α ( t ) / d t | t = = i ω α , and C α = u st α ( t ) = e i ω α t thatappears in the Heisenberg picture of a time-independent har-monic oscillator of frequency ω α , and can be expressed asfollows u α ( t ) = e i ω α t ξ α (cid:32) + ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α (cid:96) (( (cid:96) − ) ! ) cos ( (cid:96) Ω rf t ) (cid:33) , (23)where we used the secular frequencies (11) and the normal-ization (12). The operator constant of motion is built fromthe Wronskian of the position operator and the mode function (cid:99) W α ( t ) = u α ( t ) dˆ r α ( t ) / d t − ˆ r α ( t ) d u α ( t ) / d t , namely a α ( t ) = i (cid:114) M ω α (cid:18)(cid:99) W α ( t ) − (cid:90) t d t (cid:48) u α ( t (cid:48) ) F α ( t (cid:48) ) (cid:19) , (24)which fulfills a α ( t ) = a α , where a α = (cid:114) M ω α (cid:18) ˆ r α + i M ω α ˆ p α (cid:19) (25) is the standard annihilation operator of a harmonic oscillatorvibrating at the secular frequency. Using these expressions,and keeping once more the slowly-varying terms under theintegral of Eq. (23), we find that the quantum-mechanical po-sition operator can be expressed as followsˆ r α ( t ) = ˆ r sec α ( t ) + ˆ r in α ( t ) + r ex α ( t ) (cid:98) I . (26)Here, the secular-motion position operator is given byˆ r sec α ( t ) = √ M ω α ξ α (cid:16) a † α e i ω α t + a α e − i ω α t (cid:17) . (27)The intrinsic micromotion operator can be expressed asˆ r in α ( t ) = ˆ r sec α ( t ) (cid:32) ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α (cid:96) (( (cid:96) − ) ! ) cos ( (cid:96) Ω rf t ) (cid:33) , (28)and is thus again proportional to the secular motion, as oc-curred in the classical case (14). Since the secular mo-tion is expressed in terms of quantum-mechanical creation-annihilation operators, the intrinsic micromotion can be ar-gued to be a quantum-mechanical motion synchronous withthe r.f. oscillating field, as we advanced in the introduction ofthis work. As a consistency check, we note that to linear orderin q α , we recover the expressions described in [6].Finally, Eq. (26) also includes the effects of excess micro-motion in the position operator, which are proportional to theidentity operator in the vibrational Hilbert space (cid:98) I . As ex-pected from the forces in Eq. (19), the excess micromotioncorresponds to a simple displacement over the position oper-ator ˆ r sec α ( t ) → ˆ r sec α ( t ) + r driv α ( t ) (cid:98) I , the magnitude of which co-incides exactly with the classical driven amplitude (18). Onethus finds that the deviations from an ideal Paul trap affectthe quantum-mechanical position operator (26) through theclassical expression r ex α ( t ) of the excess micromotion (17).Accordingly, the excess micromotion can be considered as aclassical driven motion that can indeed be compensated byminimizing the spurious terms (2), in contrast to the intrinsicmicromotion. C. Quantum-mechanical treatment of micromotion in atrapped-ion crystal
The standard treatment of phonons in solids considers smallquantized displacements of the ions δ ˆ r i , α with respect to anunderlying Bravais lattice r i , α , namely ˆ r i , α ( t ) = r i , α + δ ˆ r i , α ,where i ∈ { , . . . , N } labels a particular ion. The collectivemodes of vibration, whose quantum-mechanical excitationslead to the aforementioned phonons, are usually obtained inthe harmonic approximation by expanding the inter-ionic po-tential to second order in the displacements, and diagonalisingthe resulting quadratic Hamiltonian [54]. For a collection of N ions inside a Paul trap, an analogous treatment exists inthe so-called pseudo-potential approximation, which assumesthat the ions are effectively trapped by a time-independentquadratic potential with secular trap frequencies (11). Themain difference with respect to a solid is that the equilib-rium positions r i α do not correspond to a Bravais lattice, butinstead form an inhomogeneous array known as a Coulombcrystal [55]. This approximation, however, does not includepossible effects of micromotion in the ion crystal.A careful classical treatment of the crystal micromo-tion [56] has recently shown that it can have non-trivial ef-fects, such as a renormalisation of the normal-mode fre-quencies in planar crystals [57]. We now present a detailedquantum-mechanical treatment of both the intrinsic and ex-cess micromotion in ion crystals, which combines the tech-niques presented in Sec. II A with the formalism in [56] to de-scribe the effect of micromotion on the classical crystal, andthen generalises Sec. II B to describe quantum-mechanicallythe effect of micromotion on the phonons of the ion crystal.To incorporate the different types of micromotion intro-duced above, we generalise the quantum-mechanical Hamil-tonian (19) to a system of N ions confined by an oscillatingquadrupole H = ∑ i , α (cid:32) ˆ p i , α M + K α ( t ) ˆ r i , α − MF i , α ( t ) ˆ r i , α (cid:33) + ∑ i , j ˜ Q | ˆ rrr i − ˆ rrr j | , (29)where we have introduced vectorial operators defined in termsof the unit cartesian vectors ˆ rrr i = ∑ α ˆ r i , α e α , and the position-momentum operators now fulfill [ ˆ r i , α , ˆ p j , β ] = i δ α , β δ i , j . Here,˜ Q = Q / πε eases notation, and we have used the springconstants (20) and time-dependent forces (21) introducedabove, allowing the spurious d.c. fields in Eq. (5) to be in-homogeneous along the crystal. The Heisenberg equations ofmotion lead to a system of equationsd ˆ r i , α d t + K α ( t ) M ˆ r i , α − ˜ Q ∑ j (cid:54) = i ˆ r i , α − ˆ r j , α | ˆ rrr i − ˆ rrr j | = F i , α ( t ) . (30)Paralleling the standard treatment of phonons in solids, wesubstitute ˆ r i , α → r i , α ( t ) (cid:98) I + δ ˆ r i , α , (31)where r i , α ( t ) are the equivalent of the equilibrium positions insolids, which become time-dependent quantities in the pres-ence of micromotion (i.e. breathing crystal), and δ ˆ r i , α arethe small quantized vibrations around such a breathing crys-tal. When these vibrations are sufficiently small, the equa-tions (30) decouple into (i) a classical system of differentialequations for the coordinates of the breathing crystal, and (ii) a linear system of equations for the quantum-mechanical dis-placements.Let us focus on (i) , and rescale the time τ = Ω rf t , such thatthe time-periodic breathing crystal fulfilsd r i , α d τ + (cid:0) a α − q α cos 2 τ (cid:1) r i , α − Q M Ω ∑ j (cid:54) = i r i , α − r j , α | rrr i − rrr j | = . (32)These differential equations correspond to a system of cou-pled Mathieu equations (3) and, inspired by the previous section, we thus propose a Floquet-type ansatz the form ofEq. (6), namely r i , α ( τ ) = ∑ n ∈ Z C α n , i ( A α e i ( β α + n ) τ + B α e − i ( β α + n ) τ ) , (33)where A α , B α are constants that depend on the initial condi-tions, β α is the so-called characteristic exponent, and C α n , i arethe Floquet coefficients. The breathing crystal corresponds toa classical solution of the type (33) synchronous with the r.f.potential, i.e. β α =
0, and can also be considered as part of theexcess micromotion due to ion positions lying off the r.f. null.By substituting this expression (33) in Eq. (32), one observesthat the Coulomb repulsion can introduce higher harmonicsof the r.f. frequency. For the linear ion crystals of interest toour purposes, these effects are absent in the relevant parame-ter regime (8), where the Floquet coefficients fulfill a systemof coupled recursion relations C α n + , i − D α n C α n , i − ∑ j (cid:54) = i ζ ( C α n , i − C α n , j ) (cid:2) ∑ α ( C α , i − C α , j ) (cid:3) + C α n − , i = . (34)Here, we have used the same notation as in the recursion rela-tion for a single ion (7), D α n = ( a α − n ) / q α , and introducedthe parameter ζ = Q / M Ω q α . To the lowest possible orderin (8), one finds that all the Floquet coefficients C α ± (cid:96), i = ( − ) (cid:96) q (cid:96) α C α , i (cid:96) (( (cid:96) − ) ! ) (35)are expressed in terms of the time-independent one C α , i . Thiscoefficient is in turn determined by the solutions of the fol-lowing system of algebraic equations M ω α C α , i − ∑ j (cid:54) = i ˜ Q ( C α , i − C α , j ) (cid:2) ∑ α ( C α , i − C α , j ) (cid:3) = , (36)where we have made use of the secular trapping frequenciesintroduced in Eq. (11). Let us note that these equations dis-play a clear competition between the harmonic trapping andthe Coulomb repulsion, and coincide with those that deter-mine the equilibrium positions of the ion crystal in the pseudo-potential approximation [55]. Therefore, we shall denote thesolutions as r eq i , α = C α , i , which can be found numerically.Since we are interested in the linear-trap configura-tion ω z (cid:28) ω x , ω y , where r eq i , α = z i δ α , z , and the oscillatingquadrupole has no effect along the trap axis of an ideal Paultrap q z =
0; we find that C α ± (cid:96), i = ∀ (cid:96) ≥ q x , q y can lead to corrections [56], but these are negli-gible in the regime of Eq. (8). Accordingly, the time-periodicbreathing crystal (33) in an ideal Paul trap corresponds to astatic Coulomb crystal rrr i ( t ) = z i e z . (37)In a segmented linear trap, where residual axial micromo-tion may exist 0 < q z (cid:28) q x , q y , one would still obtain a staticcrystal to leading order. Conversely, for crystalline solutionswhere ions lie off the trap axis, the higher-order harmonics in-troduced by the Coulomb interaction in Eq. (32) for a breath-ing crystal with q x , q y > (ii) , namely the quantum-mechanical displacements about thecrystal. After linearization, one can show that the correspond-ing operators evolve according to a system of forced Mathieuequations, similar to the single-ion case (22), but now coupledvia the linearised Coulomb interactiond d t δ ˆ r i , α ( t ) + ∑ j K α i j ( t ) M δ ˆ r j , α ( t ) = F i , α ( t ) , (38)where we have now time-dependent spring constants that cou-ple distant ions K α i j ( t ) = δ i , j K α ( t ) + V α i j ( t ) . (39)Here, we have used the single-ion spring constants (20), andintroduced the matrix of Coulomb-mediated couplings V α i j ( t ) = ( − δ i , j ) ˜ Q | rrr i ( t ) − rrr j ( t ) | ( δ α , x + δ α , y − δ α , z ) − ∑ (cid:96) (cid:54) = i δ i , j ˜ Q | rrr i ( t ) − rrr (cid:96) ( t ) | ( δ α , x + δ α , y − δ α , z ) . (40)For the linear crystals (37) that concern us in this work,this coupling matrix becomes time-independent V α i j ( t ) = V α i j ,and the system of differential equations can be decoupled bya single orthogonal transformation, in analogy to the standardtheory of phonons in solids [54]. We thus introduce the fol-lowing normal-mode operatorsˆ R m , α ( t ) = ∑ i M α i , m δ ˆ r i , α ( t ) , ˆ P m , α ( t ) = ∑ i M α i , m ˆ p i , α ( t ) , (41)where the orthogonal matrix is determined by diagonalizingthe matrix of Coulomb-mediated couplings ∑ i , j M α i , n V α i j M α j , m = δ n , m V α m , (42)where we have introduced the eigenvalues V α m . Using the or-thogonality of the transformation, we find a set of decoupledforced Mathieu equations for the normal-mode operatorsd d t ˆ R m , α ( t ) + κ α m ( t ) M ˆ R m , α ( t ) = F m , α ( t ) . (43)Here, the eigenvalues of the spring-coupling matrix κ α m ( t ) = K α ( t ) + V α m (44)inherit the time-dependence via the single-ion spring con-stants (20), and we have introduced forces that tend to displacethe ions along the normal-mode directions F m , α ( t ) = ∑ i M α i , m ˜ F i , α ( t ) . (45) Hence, we have reduced the dynamics of the small quantumdisplacements about the crystalline solution into 3 N instancesof the single-ion problem (22). We must thus find 3 N oper-ators that are constants of motion, which requires finding aset of normal mode functions u m , α ( t ) that are solutions of thehomogeneous Mathieu equations (43), namely u m , α ( t ) = ∑ n ∈ Z C α n , m ( A m , α e i ( β m , α + n ) Ω rf t + B m , α e − i ( β m , α + n ) Ω rf t ) . (46)This is the generalization of Eq. (6) with constants A m , α , B m , α that depend on the initial conditions, characteristic expo-nents for each normal mode β m , α , and Floquet coeffi-cients C α n , m . We impose the initial conditions u m , α ( ) = , d u m , α ( t ) / d t | t = = i ω m , α , and C α , m =
1, such that the modefunctions to leading order in (8) can be expressed as u m , α ( t ) = e i ω m , α t ξ α (cid:32) + ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α (cid:96) (( (cid:96) − ) ! ) cos ( (cid:96) Ω rf t ) (cid:33) , (47)with the normalization constant defined in Eq. (12), and thenormal-mode secular frequencies ω m , α = (cid:113) ω α + V α m , (48)where we have used the secular frequency in Eq. (11).Given these normal-mode functions, one can obtain theWronskian and the constants of motion through a straightfor-ward generalisation of Eq. (24) by introducing the annihilationoperators for each collective vibrational mode a m , α = (cid:114) M ω m , α (cid:18) ˆ R m , α + i M ω m , α ˆ P m , α (cid:19) . (49)Therefore, the analogue of Eq. (26) for the quantum-mechanical treatment of micromotion in a trapped-ion crystalcan be expressed asˆ r i , α ( t ) = r i , α ( t ) (cid:98) I + δ ˆ r sec i , α ( t ) + δ ˆ r in i , α ( t ) + r ex i , α ( t ) (cid:98) I . (50)Here, the secular-motion position operator is given by δ ˆ r sec i , α ( t ) = ∑ m M α i , m (cid:112) M ω m , α ξ α (cid:16) a † m , α e i ω m , α t + a m , α e i ω m , α t (cid:17) , (51)and the intrinsic micromotion operator can be expressed as δ ˆ r in i , α ( t ) = δ ˆ r sec i , α ( t ) (cid:32) ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α (cid:96) (( (cid:96) − ) ! ) cos ( (cid:96) Ω rf t ) (cid:33) . (52)The excess micromotion in Eq. (50) is expressed in terms ofthe identity operator in the vibrational Hilbert space (cid:98) I , and r ex i , α ( t ) = r driv i , α ( t ) (cid:32) + ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α (cid:96) (( (cid:96) − ) ! ) cos ( (cid:96) Ω rf t ) (cid:33) , (53)where we have introduced the generic site-dependent ampli-tude r driv i , α ( t ) . For the standard Paul trap, this can be obtainedfrom Eq. (18) by considering inhomogeneous spurious fields r driv i , α ( t ) = QE α dc , i M ω α + δ α , x q x r ϕ ac ˜ α ( Ω rf t ) . (54)For other situations, such as those arising for segmented traps,this amplitude will depend on the specific trap details.In this way, we have presented a detailed quantum-mechanical description of the effects of intrinsic and excessmicromotion in a linear crystal of trapped ions. The resultsin Eqs. (50)-(54) will be the starting point for the scheme ofmicromotion-enabled entangling gates in the following sec-tion. Our formalism can be extended to planar crystals, al-though one has to consider the breathing of the crystal insteadof Eq. (37), and how this can lead to micromotion-inducedcorrections of the secular vibrations of the planar crystal. III. ENTANGLING GATES BASED ON MICROMOTIONSIDEBANDS
In this section, we start in III A by discussing the effectsof micromotion in the theory of light-matter interactions fora set of laser beams addressed to a particular electronic tran-sition of the ions. We then describe how to implement state-dependent forces by combining pairs of laser beams in III B, and discuss the role of intrinsic/excess micromotion, payingspecial attention to the contribution of the often-neglected car-rier excitations. In III C, we start by reviewing the schemesfor entangling gates based on secular state-dependent σ φ -forces [29, 31–34], and discussing the gate speed limitationsthat arise due to the off-resonant carrier. Building on this dis-cussion, we then introduce a scheme of micromotion-enabledstate-dependent σ φ -forces, which can overcome the limita-tions on the gate speed due to the off-resonant carrier, pro-vided that the excess micromotion is accurately compensated. A. Micromotion effects in the laser-ion interaction
Let us consider a collection of N trapped ions subjected tolaser beams tuned close to the resonance of a particular tran-sition of frequency ω between two electronic levels |↑ i (cid:105) , |↓ i (cid:105) .The dynamics of the internal and motional degrees of freedomof this ion crystal is described by the following Hamiltonian H ( t ) = ∑ i ω σ zi + ∑ i , α (cid:32) ˆ p i , α M + K α ( t ) δ ˆ r i , α − MF i , α ( t ) δ ˆ r i , α + ∑ j V α i j δ ˆ r i , α δ ˆ r j , α (cid:33) , (55)where we have introduced σ zi = |↑ i (cid:105) (cid:104)↑ i | − |↓ i (cid:105) (cid:104)↓ i | , appliedthe harmonic-crystal approximation described in the previoussection (31), and neglected an irrelevant c -number stemmingfrom the classical energy of the breathing crystal. The inter-action between the laser beams and the ions is described by H I ( t ) = ∑ i , l Ω l i φ l σ + i e i ( kkk l · ˆ rrr i ( t )+( ω − ω l ) t ) + H . c ., (56)where we have introduced the spin raising σ + i = |↑ i (cid:105) (cid:104)↓ i | andlowering σ − i = |↓ i (cid:105) (cid:104)↑ i | operators. Here, l labels the dif-ferent laser beams that are described as classical travelingwaves with kkk l , ω l , φ l being the laser wavevector, frequency,and phase, respectively, and Ω l is the Rabi frequency of the particular transition. Typically, one either considers aquadrupole-allowed transition between a groundstate level |↓(cid:105) and a metastable excited level |↑(cid:105) , or uses a two-photon Ra-man scheme to couple a pair of groundstate levels |↓(cid:105) , |↑(cid:105) viaa excited level through a far-off-resonant dipole transition.In any case, the quadrupole or Raman Rabi frequencies areconstrained to | Ω l | (cid:28) ω + ω l in order to neglect additionalcounter-rotating terms in Eq. (56).We note that this expression is obtained in the interac-tion picture of the bare Hamiltonian (55), namely H I ( t ) = U †0 ( t ) H I U ( t ) , where U ( t ) = T { exp ( − i (cid:82) t d t (cid:48) H ( t (cid:48) )) } . Thus,after substituting the position operator in Eqs. (50)-(54) cor-responding to such an interaction picture, we find H I ( t ) = ∑ i , l , α Ω α l , i σ + i (cid:18) + i ∑ m M α i , m η α l , m (cid:0) a † m , α u m , α ( t ) + a m , α u ∗ m , α ( t ) (cid:1) + · · · (cid:19) e i φ l , i ( t ) e i ( ω − ω l ) t + H . c ., (57)where we have performed a Taylor series in the Lamb-Dicke parameters η α l , m = kkk l · e α / (cid:112) M ω m , α (cid:28) Ω α l , i = Ω l exp {− ∑ m ( M α i , m η α l , m ) / } , together with laserphases that get modulated by the time-dependence of the breathing crystal and the excess micromotion φ l , i ( t ) = φ l + kkk l · ( rrr i ( t ) + rrr ex i ( t )) . (58)Note that for a linear chain, the breathing crystal becomesstatic rrr i ( t ) = z i e z (37), such that the phase modulation is onlycaused by the excess micromotion. Additionally, the effectof the intrinsic micromotion on the laser-ion interaction is en-coded in the particular time-dependence of the mode functions u m , α ( t ) (47), which yield additional periodic modulations inprocesses involving the creation and annihilation of phonons.Let us note that the mode functions are written in Eq. (47)as u m , α ( t ) = e i ω m , α t f α in ( t ) , were f α in ( t ) is a function with period2 π / Ω rf already written as a Fourier series. The excess micro-motion leads to f l ex ( t ) = e i φ l , i ( t ) , which is also a periodic func-tion with period 2 π / Ω rf , and could as well be expressed as aFourier series with all the possible harmonics at the differentfrequencies (cid:96) Ω rf . In this sense, the micromotion introduces acomb of equidistant sidebands in the laser-ion interaction (57),the so-called micromotion sidebands . These can be exploitedby choosing an appropriate detuning of the lasers with respectto the atomic transition ω l − ω ≈ (cid:96) (cid:63) Ω rf , (59)where (cid:96) (cid:63) ∈ Z is a certain integer. In most trapped-ion ex-periments, one sets (cid:96) (cid:63) = (cid:96) (cid:63) =
1, while maintaining the compensa-tion of excess micromotion, can be advantageous for QIP. Inthis way, one can exploit the effects of intrinsic micromotionin the laser-ion interaction, and find faster and more accurateschemes for entangling quantum logic gates.
B. State-dependent dipole forces and off-resonant carriers
We now discuss how to induce a state-dependent σ φ -force [29] on the ions starting from Eq. (57), and thus tak-ing into account the new effects brought forth by micromo-tion. We consider a pair of laser beams l ∈ { , } with equalwavevectors kkk = kkk : = kkk L . By selecting the direction of thesebeams along a certain trap axis kkk L || e α , the laser-ion interac-tion will only couple the qubits to a particular phonon branch.We also consider equal laser phases φ = φ = : φ , and equalintensities and polarizations leading to Ω = Ω = : Ω . Con-versely, the lasers will have opposite detunings with respectto the atomic transition δ : = ω − ω = − ( ω − ω ) . Due tothese choices, we can simplify the laser-ion interaction (57)considerably by defining common Lamb-Dicke parameters η α , n = η α , n = : η α n , dressed Rabi frequencies Ω α , i = Ω α , i = : Ω α i , and modulated phases φ , i ( t ) = φ , i ( t ) = : φ i ( t ) , where φ i ( t ) = φ + k L r i , α + k L r driv i , α ( t ) (cid:32) + ∑ (cid:96) ≥ ( − ) (cid:96) q (cid:96) α cos ( (cid:96) Ω rf t ) (cid:96) (( (cid:96) − ) ! ) (cid:33) . (60)By keeping contributions to first order in the Lamb-Dicke pa-rameter, H I ( t ) = H c ( t ) + H s ( t ) , we identify the terms drivingthe carrier transitions H c ( t ) = ∑ i Ω α i σ + i e i φ i ( t ) cos δ t + H . c ., (61) and the spin-phonon couplings H s ( t ) = ∑ i , m i F α i , m x α m (cid:0) a † m , α u m , α ( t ) + H . c . (cid:1) σ + i e i φ i ( t ) cos δ t + H . c ., (62)where the dipole forces are F α i , m = Ω α i M α i , m k L , and the n -thmode groundstate widths are x α m = / (cid:112) M ω m , α .We now consider the effects of the excess micromotion (60)to leading order in the regime (8), namely φ i ( t ) ≈ ϕ i + ˜ β i cos ( Ω rf t ) , (63)where we have introduced the parameters ϕ i = φ + k L ( r i , α + r driv i , α ( )) , ˜ β i = − k L r driv i , α ( ) q α . (64)Using the Jacobi-Anger expansion [58], one findse i φ i ( t ) = e i ϕ i ∑ (cid:96) ∈ Z J (cid:96) ( ˜ β i ) e i ( (cid:96) π + (cid:96) Ω rf t ) , (65)where J (cid:96) ( x ) are the (cid:96) -th order Bessel functions of the firstclass. This is the explicit expression for the Fourier series thatwas mentioned above Eq. (59), and leads to a clear picture forthe appearance of the micromotion sidebands at frequencies (cid:96) Ω rf . Depending on the particular value of the laser detun-ing δ ≈ (cid:96) (cid:63) Ω rf , it is possible to address a particular micromo-tion sideband (59). Moreover, around each of these micro-motion sidebands, there is an additional comb of frequenciesrepresenting the secular sidebands that occur at multiples ofthe secular normal-mode frequencies (48). By combining apair of first secular sidebands, the spin-phonon couplings ofEq. (62) yield the desired state-dependent force.
1. Secular state-dependent dipole forces
The usual approach to obtain a state-dependent force relieson addressing the secular sidebands, δ ∼ ω α (cid:28) Ω rf [29, 31–34], such that (cid:96) (cid:63) = | Ω α i | (cid:28) Ω rf , (66)and using the expression in Eq. (65) for the effects of excessmicromotion, and Eq. (47) for the effects of the intrinsic mi-cromotion, we find that the secular sidebands (62) can be ex-pressed as a Hamiltonian with a state-dependent force H s ( t ) ≈ ∑ i , m F r i , m x α m s i a † m , α e i ω m , α t cos δ t + H . c ., (67)where we have introduced a dipole-force strength F r i , m = Ω α i M α i , m k L ( − q α ) (cid:114) J ( ˜ β i ) + (cid:16) q α J ( ˜ β i ) (cid:17) , (68)and the following spin operator s i = (cid:113) J ( ˜ β i ) + q α J ( ˜ β i ) (cid:16) J ( ˜ β i ) ˜ σ yi + q α J ( ˜ β i ) ˜ σ xi (cid:17) . (69)0Here, we have defined the Pauli matrices in a rotated basiswith respect to the z -axis˜ σ xi : = e i ϕ i σ zi ( σ + i + σ − i ) e − i ϕ i σ zi , ˜ σ yi : = e i ϕ i σ zi ( i σ − i − i σ + i ) e − i ϕ i σ zi . (70)Accordingly, the spin operator (69) shares certain algebraicproperties with the rotated Pauli matrices in Eq. (70), namely s i = s † i , s i = I , and [ s i , s j ] =
0, which allow us to interpretEq. (67) as a state-dependent force that pushes the vibrationalmodes in opposite directions depending on the two eigenstatesof the spin operator s i = | + s i (cid:105) (cid:104) + s i | − |− s i (cid:105) (cid:104)− s i | . In the limitof vanishing excess micromotion β i = ϕ i ≈ φ + k L z i δ α , z , and Eq. (67) yields the aforementioned σ φ -forceof the Mølmer-Sørensen (MS) scheme [29] used in severalexperiments [31–34], where σ φ = ie i φ σ + − ie − i φ σ − .Let us note that, in addition to the desired state-dependentforce (67), one has to consider the carrier terms (61), which inthis regime (66) can be expressed as H c ( t ) = ∑ i Ω α i J ( ˜ β i ) ˜ σ xi cos δ t . (71)This residual carrier does not commute with the dipoleforce (67), since the rotated Pauli matrices share the same su ( ) algebra as the original Pauli matrices. Therefore, thecarrier and the dipole force will interfere and compromise thesimple picture of the normal modes being displaced in oppo-site directions depending on the spin state. To minimise thisundesired effect, the residual carrier must be far off-resonant,which can be achieved by limiting the laser intensity such that | Ω α i J ( ˜ β i ) | (cid:28) δ ∼ ω α , (72)and H c ( t ) ≈ β i →
0, this constraint (72)reduces to the standard condition required to work in theresolved-sideband regime | Ω α i | (cid:28) ω α . As a consequence, re-solving the secular sidebands limits the intensity of the state-dependent force (68) that becomes in this regime F r i , m = Ω α i M α i , m k L ( − q α ) , ˜ s i ≈ ˜ σ yi (73)As discussed in the following section, it puts a constraint onthe speed of entangling gates based on σ φ -forces. Hence, itwould be desirable to come up with schemes that yield sim-ilar state-dependent forces with milder constraints on theirstrengths. We now argue that this is possible by exploitingthe higher micromotion sidebands.
2. Micromotion state-dependent dipole forces
Let us now discuss how to obtain a state-dependent forceby addressing the first micromotion sideband, δ = Ω rf + ˜ δ ,where ˜ δ ∼ ω α (cid:28) Ω rf , such that (cid:96) (cid:63) = H s ( t ) ≈ ∑ i , m ˜ F r i , m x α m ˜ s i a † m , α e i ω m , α t cos ˜ δ t + H . c ., (74)where we have introduced a dipole-force strength˜ F r i , m = Ω α i M α i , m k L ( − q α ) (cid:114) J ( ˜ β i ) + (cid:16) q α J ( ˜ β i ) (cid:17) , (75)and the following spin operator˜ s i = (cid:113) J ( ˜ β i ) + q α J ( ˜ β i ) (cid:16) − J ( ˜ β i ) ˜ σ xi + q α J ( ˜ β i ) ˜ σ yi (cid:17) . (76)In analogy with the secular forces (67), we can interpretEq. (74) as a state-dependent force that pushes the vibrationalmodes in opposite directions depending on the two eigenstatesof the spin operator ˜ s i = | + ˜ s i (cid:105) (cid:104) + ˜ s i | − |− ˜ s i (cid:105) (cid:104)− ˜ s i | .The additional carrier term in Eq. (61) can be expressed inthis case as H c ( t ) ≈ ∑ i (cid:16) Ω α i J ( ˜ β i ) ˜ σ xi cos Ω rf t − Ω α i J ( ˜ β i ) ˜ σ yi cos ˜ δ t (cid:17) . (77)In principle, this term can cause a similar interference with thestate-dependent force (74), since it does not commute with thespin operator in general (76). However, if the excess micro-motion is minimized to the level˜ β i (cid:28) q α (cid:28) , (78)one gets J ( ˜ β i ) ≈ J i ( ˜ β i ) ≈ ˜ β i , such that the previouscondition (72) to neglect the off-resonant carrier becomes lessstringent. We find that the laser intensity will be limited by | Ω α i | (cid:28) Ω rf , | Ω α i | ˜ β i (cid:28) ˜ δ ∼ ω α , (79)and can be thus tuned to larger values in comparison to thesecular scheme (72), where | Ω α i | (cid:28) ω α . According to thisdiscussion, the advantage of the micromotion-enabled schemein minimising the undesired effects brought up by the off-resonant carrier with respect to the standard secular schemewill be larger the smaller ω α / Ω rf and ˜ β i can be made in theexperiment. This will depend on the particular trap architec-ture, and the excess micromotion compensation capabilitiesdiscussed below. Let us finally note that the state-dependentforce (75) becomes in this regime˜ F r i , m ≈ Ω α i M α i , m k L q α ( − q α ) , ˜ s i ≈ ˜ σ yi . (80)Comparing the strength of the secular (73) and micromo-tion (80) forces, one can see that to obtain similar strengthsone would need to increase the Rabi frequency in themicromotion-scheme, and thus the laser power, by a factorof roughly 4 / q α . At this point, it is worth noting that wecould have tuned the laser frequencies to a higher micromo-tion sideband δ = Ω rf + (cid:96) (cid:63) ˜ δ with (cid:96) (cid:63) >
1. By doing this, the1effect of the off-resonant carrier would be further suppressed | Ω α i | (cid:28) (cid:96) (cid:63) Ω rf . On the other hand, we would need even higherlaser powers, increased by a factor of 4 (cid:96) (cid:63) (( (cid:96) (cid:63) − ) ! ) / ( q α ) (cid:96) (cid:63) ,to achieve forces of the same strength. Even if these laser in-tensities can be achieved in the laboratory, in this regime theintensity fluctuations could become a limiting factor for thegate performance. Accordingly, we will focus on the first mi-cromotion sideband in the rest of this work. C. Entanglement via geometric phase gates
We now discuss how to exploit the longitudinal/transversephonons to mediate a qubit-qubit interaction capable of gener-ating entanglement in the presence of micromotion. In orderto have a simple description, we make use of the Magnus ex-pansion [63], which allows us to express the time-evolutionoperator in the interaction picture as follows U I ( t ) = T (cid:110) e − i (cid:82) t d t (cid:48) H I ( t (cid:48) ) (cid:111) = e A ( t ) . (81) Here, the anti-Hermitian operator A ( t ) = − A † ( t ) can be ex-pressed as a series of time integrals over nested commutators A ( t ) = − i (cid:90) t d t H I ( t ) − (cid:90) t d t (cid:90) t d t [ H I ( t ) , H I ( t )] + · · · , (82)which can be truncated to the desired order of approxima-tion. This will allow us to discuss the generation of entangle-ment through a generic Hamiltonian with a state-dependentforce H I ( t ) = H s ( t ) , which encompasses both Eq. (67) andEq. (74), and allows for an additional pulse shaping on theforces F r i , m → F r i , m ( t ) .In this ideal situation, the Magnus expansion (82) becomesexact already at second order, such that A ( t ) = ∑ i , m s i (cid:0) γ i , m ( t ) a m , α − γ ∗ i , m ( t ) a † m , α (cid:1) + ∑ i , j g i j ( t ) s i s j , (83)where we have introduced the following parameters γ i , m ( t ) = − i (cid:90) t d t F r i , m ( t ) x α m cos ( δ t ) e − i ω m , α t , (84) g i j ( t ) = i (cid:90) t d t (cid:90) t d t ∑ m F r i , m ( t ) x α m F r j , m ( t ) x α m cos ( δ t ) cos ( δ t ) sin (cid:0) ω m , α ( t − t ) (cid:1) , (85)Hence, the Magnus expansion operator (83) amounts to astate-dependent displacement of the vibrational modes, fol-lowed by an effective spin-spin interaction that is capableof generating the desired entanglement between the trapped-ion qubits. On the contrary, the displacement will degradethe quality of the quantum logic gate, as it leads to resid-ual entanglement between the qubits and the phonons, con-tributing with a motional error that must be minimised. If γ i , m ( t g ) ≈
0, the vibrational modes develop a closed trajectoryin phase space, returning to the initial state after a particulargate time t g . Along these closed trajectories, the qubits ac-quire a state-dependent geometric phase that depends on theenclosed phase-space area, and can be exploited to generatemaximally entangled states [27, 29].For instance, considering N = ρ = | ψ (cid:105) (cid:104) ψ | ⊗ ρ th , where | ψ (cid:105) = |↓ ↓ (cid:105) is the state of two qubitsafter optical pumping, and ρ th is the state of the vibrationalmodes after laser cooling, the time-evolved state under a sec-ular state-dependent force (67) in the limit of β i (cid:28) ρ ( t g ) = | ψ ( t g ) (cid:105) (cid:104) ψ ( t g ) | ⊗ ρ th , where | ψ ( t g ) (cid:105) = √ |↓ ↓ (cid:105) + i √ i ( ϕ + ϕ ) |↑ ↑ (cid:105) (86)is locally equivalent to a Bell state, and we have assumed thatthe laser intensities have values such that 2 g ( t g ) = − i π / γ i , m ( t g ) ≈
0, and g i j ( t g ) = − i π /
1. Entangling gates with secular forces
Let us particularize the Magnus operator (83) to the secularstate-dependent dipole force (67), which we will assume to be2composed of a sequence of N p square pulses F r i , m ( t ) = N p ∑ n p = f n p i , m (cid:0) θ ( t − t n p ) − θ ( t − ( t n p + τ n p )) (cid:1) . (87)Here, f n p i , m is the force of the n p -th pulse obtained from Eq. (68)by substituting the Rabi frequency Ω α i , n p of that particularpulse, and θ ( x ) is the Heaviside step function, such that thispulse acts within a time window t ∈ [ t n p , t n p + τ n p ) . In order touse the above generic Magnus expansion, the additional off-resonant carrier (77) must be negligible, which requires thelaser parameters to lie in the regime (72). Moreover, we willfocus on the regime where the excess micromotion is very-well compensated, such that Eq. (72) leads to | Ω i | (cid:28) ω α , andthus | f n p i , m x m | (cid:28) ( δ + ω m ) follows from Eq. (68). This con-straint over the forces allows us to simplify considerably theparticular expressions for the parameters (84) and (85) forsingle- and multi-pulse gates. (i) Single-pulse entangling gates: Let us consider Eq. (87)for a single pulse n p = N p = f i , m , between t n p = τ n p = t g [29, 31–34]. By performing the correspondingintegrals, one finds the state-dependent displacements γ i , m ( t g ) ≈ f i , m x α m − e i ( δ − ω m , α ) t g δ − ω m , α , (88)and the phonon-mediated spin-spin interactions g ( t g ) = i ∑ m f , m x α m f , m x α m ( ω m , α − δ ) (cid:18) ω m , α t g + ω m , α sin ( δ − ω m ) t g δ − ω m , α (cid:19) . (89) (a) Addressing a single vibrational mode: Let us start byconsidering single-pulse gates that resolve a single bus modeto mediate the interaction, such as the center-of-mass (CoM)mode δ ≈ ω , α of either longitudinal or transverse vibrations.The condition to resolve a single vibrational mode for a N = | f i , m x α m | (cid:28) | ω , α − ω , α | , (90)such that γ i , ( t g ) ≈ t g such that γ i , ( t g ) = t g = π r | δ − ω , α | , (91)where r ∈ Z + [29]. The phase-space trajectory defined by γ i , ( t g ) =
0, and induced by the displacement operator (83),corresponds to r closed circular loops, such that spin and mo-tional degrees of freedom get disentangled at the end of thegate. Conversely, the two spins can get maximally entangled.Using 2 g ( t g ) ≈ − i t g J , one finds that the time-evolution op-erator (81)-(83) can be expressed as U I ( t g ) = e − i t g J s s , (92) where we have introduced the spin-spin coupling strengths J = ω R Ω α Ω α J ( β ) J ( β )( − q α ) ∑ m M α , m M α , m δ − ω m , α , (93)where ω R = k / M is the recoil energy. Considering a neg-ligible excess micromotion β i (cid:28)
1, the coupling becomes J ≈ ( Ω α η α ) ω , α / ( δ − ω , α ) ≈ ( Ω α η α ) / ( δ − ω , α ) to leading order of the Lamb-Dicke parameter. This coincideswith the expression in Ref. [29] up to a different definitionof their Lamb-Dicke parameter that incorporates the normal-mode displacements.The condition to generate a maximally-entangled state us-ing Eq. (92) is J t g = π /
4, which sets another constraint be-tween laser detuning and the gate time ( Ω α η α ) | δ − ω , α | t g = π . (94)Solving the system of algebraic equations (91) and (94) fixesthe detuning as a function of the number r of closed loopsin phase space, and the Rabi frequency of the transition Ω α .Accordingly, the gate time can be shown to be t g = π Ω α η α (cid:112) r , (95)such that the stronger the intensity of the laser is, the larger Ω α becomes, and the faster the entangling gate is, e.g. t g = √ π / Ω α η α for gates based on 1-loop trajectories. We notethat this intensity increase must be accompanied by a corre-sponding increase in the detuning of the laser beams δ = ω α + (cid:112) r Ω α η α . (96)However, such an increase in gate speed cannot be prolongedindefinitely. Let us recall that the condition to resolve a singlevibrational mode (90) puts a constraint on the laser intensity Ω η α (cid:28) | ω , α − ω , α | , such that the gate speed is limited by t g (cid:29) π | ω , α − ω , α | . (97)This gate-speed limitation is very different for MS gates thatuse longitudinal or transverse phonons as the quantum bus tomediate the qubit-qubit entanglement. (a.1) For longitudinal phonons , the modes fulfill | ω , z − ω , z | ∼ ω z , such that the gate speed is ultimately limited bythe trap period t g (cid:29) π / ω z . Let us emphasize, however, thatthe gate fidelity would decrease for such fast gates, which setsa lower speed limit in practice. Maximising the gate speedby increasing the Rabi frequency within the valid parameterregime (90), namely Ω z (cid:28) | ω , z − ω , z | / η z ∼ ω z / η z , can leadto situations where the contribution of the off-resonant car-rier (77) is not negligible anymore, i.e. Ω z (cid:28) ω z in Eq. (72)begins to be violated. Accordingly, if the gate speed increasesbeyond a certain limit, the off-resonant carrier will increasethe gate error and dominate over other sources of noise.To quantify this effect, we estimate the state infidelity ε g = − F g for the generation of the desired Bell state (86)3 − T = . s T = . s T = . s T = . s e mot e d e p h e ca rr e d e p h e d e ph e d e ph t g ( µ s )
300 35050 1000 e g / Figure 1.
Single-pulse MS gate with the axial CoM mode:
Stateinfidelity ε g for a MS gate mediated by the CoM longitudinal modeof a N = Ca + ions. We assume an axial trap frequency ω z / π = . η z = . n z = . r =
1. The blue solid lines correspond to the total state in-fidelity ε g for dephasing times T ∈ { . , . , . , . } s , whereas thedotted lines represent the contributions of dephasing, motional, andcarrier errors, as indicated in the captions. The yellow stars representthe optimum gate times with respect to the highest-possible gate fi-delity for each set of parameters. To vary the gate speed, we considerincreasing the Rabi frequency within Ω z / π ∈ [ . , . ] MHz, andsetting the corresponding detunings ( δ − ω z ) / π ∈ [ . , . ] kHzaccording to the equations discussed in the text. as a function of the gate time. We consider three differentsources of infidelity ε g = ε carr + ε mot + ε deph : the off-resonantcarrier (77) leads to ε carr ≈ N ( Ω z / δ ) [29, 30], the additionalterms neglected in the Lamb-Dicke expansion (57), includingthe effect of spectator modes, lead to a motional error ε mot ≈ . π N ( δ − ω z )( ¯ n z + ) / ( ω z t g ) + π N ( N − )( η z ) ( . n z + . n z ) / N [29], where we have assumed a thermal statefor the longitudinal vibrational modes, such that ¯ n z is themean number of phonons in the thermal CoM mode. Fi-nally, we also consider dephasing during the gate, which canbe caused by fluctuating global magnetic fields, which leadto ε deph ≈ N t g / T , where T is the dephasing time of thequbits, as measured by Ramsey interferometry. In Fig. 1, werepresent the full error as a function of the gate time for dif-ferent dephasing rates, choosing Ca + qubits as a represen-tative case [64]. This figure demonstrates that for cold crys-tals with ¯ n z = .
1, the motional error of MS gates is negli-gible in comparison to the errors due to the dephasing andthe off-resonant carrier. This would occur also for warmercrystals with the same parameters, provided that ¯ n z ≤
5, afterwhich the motional-error contribution cannot be neglected anylonger. Whereas for slow gates, the ε deph contribution is dom-inant, ε carr becomes the leading source of infidelity when thegate becomes sufficiently fast. As predicted above, the gate isalways slower than the trap period T t = π / ω z = µ s if oneaims for reasonably-high fidelities ε g < − . In Sec. III C 2, − T = . s T = . s T = . s T = . s e m o t e d e p h e ca rr e d e p h e d e ph e d e ph t g ( µ s )
300 35050 1000 e g / Figure 2.
Single-pulse MS gate with the transverse CoM mode:
State infidelity ε g for a MS gate mediated by the CoM transversemode of a N = Ca + ions. We assume an axial (ra-dial) trap frequency ω z / π = . ω x / π = . η x = . n x = .
05 for the CoM mode, andset the number of phase-space loops of the MS gate to r =
1. Theblue solid lines correspond to the total state infidelity ε g for dephas-ing times T ∈ { . , . , . , . } s , whereas the dotted lines representthe contributions of dephasing, motional, and carrier errors, as indi-cated in the captions. The yellow stars represent the optimum gatetimes with respect to the highest-possible gate fidelity for each set ofparameters. To vary the gate speed, we consider increasing the Rabifrequency within Ω x / π ∈ [ . , . ] MHz, and setting the corre-sponding detunings ( δ − ω x ) / π ∈ [ . , . ] kHz according to theequations discussed in the text. we will discuss how it is possible to increase the gate speedfurther, while maintaining high fidelities, provided that the in-trinsic axial micromotion of the ion crystal can be exploitedto shape the micromotion state-dependent forces (74) insteadof the secular ones (67). (a.2) For transverse phonons , the situation can first appearfavorable, since the trap frequencies are larger, and one cannaively expect that δ ∼ ω x can be achieved with larger de-tunings, and thus shorter gate times (91). However, the con-dition to resolve a single mode (90) is more stringent, since | ω , x − ω , x | ∼ ( ω z / ω x ) ω x (cid:28) ω z for the usual regime of lin-ear Paul traps ω z (cid:28) ω x . Therefore, exploiting the availablelarger detunings to speed up the gate leads inevitably to a de-crease in fidelity. We note that the condition to resolve a singlemode imposes Ω x (cid:28) | ω , x − ω , x | / η x (cid:28) ω z / η x (cid:28) ω x / η , x .Hence, even if the gate speed is maximised, one would notreach the regime where the off-resonant carrier starts to beproblematic since Ω x (cid:28) ω x is always warranted. Hence, theerror for fast MS gates will be dominated by the contributionto the motional error of the spectator modes.To quantify this discussion, we estimate again the state infi-delity ε g = − F g for generating the desired Bell state (86) asa function of the gate time. The carrier and dephasing errorshave the same expressions as above, whereas the motional er-ror changes due to the proximity of the spectator modes infrequency space. For N =
2, we get ε mot ≈ ( Ω x η x ) ( n x + )( δ + ω , x ) / ( δ − ω , x ) , where we have assumed a thermalstate for the transverse vibrational modes with mean phononnumber ¯ n x . In Fig. 2, we represent the full error as a func-tion of the gate time for different dephasing rates, choosing Ca + qubits to compare with the previous longitudinal gate.As announced earlier, this figure shows that the error of slow(fast) gates is dominated by the the dephasing (motional) er-ror. One also observes, that the optimum transverse gates arealways slower than the longitudinal ones in Fig. 1 and, more-over, achieve smaller fidelities.Let us also note that both of these longitudinal andtransverse entangling gates can be generalised to N -qubits, and would lead to multi-partite maximally-entangledstates locally-equivalent to | GHZ (cid:105) N = ( |↓ ↓ · · · ↓ N (cid:105) + |↑ ↑ · · · ↑ N (cid:105) ) / √
2, instead of the Bell state (86). The condi-tions to generate such states using MS gates based on the lon-gitudinal CoM mode remain the same, since such a bus modeis always separated from higher-frequency modes by the samefrequency gap [55]. On the contrary, the conditions on MSgates based on the transverse CoM mode lead to even slowergates, since the phonon branch becomes denser, and the dif-ferent modes approach the CoM frequency as N increases. (b) Addressing both vibrational modes: Let us now addresshow to increase the gate speed by lifting the constraint (90),such that the state-dependent force does not resolve a singlevibrational mode even when δ ≈ ω , α . For N = γ i , ( t g ) =
0, and γ i , ( t g ) =
0. The first one setsthe relation in Eq. (91) between the gate time and the detun-ing, whereas the yields a commensurability condition δ − ω , α = r ( δ − ω , α ) , (98)where r ∈ Z , which already fixes the detuning to δ = ( r ω , α − ω , α ) / ( r − ) . (99) (b.1) For longitudinal modes , the condition (98) cannot bemet, as the frequency difference is an irrational number ω , z − ω , z = ( √ − ) ω z . In any case, the gate-speed could not beincreased even if one could close both trajectories perfectly,as the limitation on gate speed is given by the condition toneglect the off-resonant carrier (see Fig. 1). Equivalently, thiswould not improve the gate fidelity of the MS gates too much,as the motional error due to the spectator vibrational mode isalready very small for typical experimental values (see Fig. 1). (b.2) For transverse modes , in contrast, the condition (98)can be met and the allowed gate times correspond to trajecto-ries with r loops for the center-of-mass mode ω , x , and r | r | for the zigzag mode ω , x with r ≥ r ≤ −
1. These twoconditions suffice to fix the gate time to t g = π r | r − | ( ω , x − ω , x ) . (100)The remaining task is to find the required laser intensitysuch that the state-dependent geometric phase proportionalto the enclosed phase-space area fulfills the condition togenerate a maximally-entangled state J t g = π /
4. In this − t g ( µ s )
300 35050 1000 e g / T = . s T = . s T = . s T = . s e mot e d e p h e ca rr e d e p h e d e ph e d e ph
412 2500
Figure 3.
Single-pulse MS gate with both transverse modes:
State infidelity ε g for a MS gate mediated by both transverse modesof a N = Ca + ions. We assume a radial trap fre-quency ω x / π = . η x = . n x = .
05 for the CoM mode, and set the number of phase-space loops of the MS gate to r =
1, and r =
2. The blue solidlines correspond to the total state infidelity ε g for dephasing times T ∈ { . , . , . , . } s , whereas the dotted lines represent the con-tributions of dephasing, motional, and carrier errors, as indicated inthe captions. The yellow stars represent the optimum gate times withrespect to the highest fidelity for each set of parameters. To vary thegate speed, we consider increasing the axial trap frequency within ω z / π ∈ [ . , . ] MHz, and setting the corresponding Rabi fre-quencies Ω x / π ∈ [ . , . ] MHz and detunings ( δ − ω x ) / π ∈ [ . , . ] kHz according to the equations discussed in the text. case, one has to consider the contribution of both modes tothe spin-spin coupling strength (93), which becomes J ≈ ( Ω x η x ) ( / ( δ − ω , x ) − / ( δ − ω , x )) . Using the expres-sion for the fixed detuning (99), one finds that the requiredlaser Rabi frequency is Ω x η x = ( ω , x − ω , x ) | r − | (cid:115) | r | r | r − | . (101)As occurred for the MS gates that use a single vibrational busmode (95), the gate can become faster by increasing the laserRabi frequency, since the expression t g = π Ω x η x (cid:115) | r | r | r − | , (102)yields t g = π / Ω x η x for the fastest gate with r = r = t g = πω , x − ω , x , (103)which shows that by exploiting both vibrational modes simul-taneously, the speed can be increased with respect to the limi-tation of the previous transverse MS gates (97).5We note that the procedure of increasing the gate speedis slightly more involved than that of single-mode MSgates (95), which only required increasing simultaneously theRabi frequency and the detuning of the laser beams (96). Fortwo-mode MS gates, Eq. (101) shows that in addition oneneeds to increase the frequency difference between both vi-brational modes, which requires modifying the trap confine-ment. In particular, we consider increasing the axial trap fre-quency ω z , since ( ω , x − ω , x ) ∼ ( ω z / ω x ) ω x , and this willincrease the gate speed (103). The ultimate limit to such anincrease in gate speed is caused by the structural instabilityof the ion chain towards a zig-zag ladder, which occurs for ω z ≈ ω x for N = T t = π / ω x . However, note that the required Rabi frequencyin this ultimate limit would largely violate the condition to ne-glect the off-resonant carrier (72), as Ω x ∼ ( ω , x − ω , x ) / η x ∼ ( ω z / ω x ) ω x / η x ∼ ω x / η x (cid:29) ω x . Accordingly, this fast gatewould have poor fidelities. Another effect that would decreasethe gate fidelity even further is the increasing importance ofnon-linear quartic terms in the vibrational Hamiltonian as oneapproaches the structural instability, which would modify thesimple phase-space trajectories of the MS schemes.Therefore, at a practical level, the limit on gate speed forhigh-fidelity MS gates based on two transverse bus modeswould be to consider ω z / ω x ∼ (cid:112) η x /
10, such that t g ≈ π / ( ω z / ω x ) ω x (cid:29) π / ω z (cid:29) π / ω x . Although this gate isstill considerably slower than the trap period, there will beparticular ratios ω z / ω x , such that the transverse MS gate maysurpass the speed of the longitudinal one. In this sense, by re-solving the two vibrational modes, the transverse MS gate canexploit the larger available detunings to achieve higher speeds,while maintaining high fidelities.To quantify this discussion, we estimate again the state infi-delity ε g = − F g for generating the desired Bell state (86) asa function of the gate time. The carrier and dephasing errorshave the same expressions as above, whereas the motional er-ror changes once more since both modes are active buses, andthe leading order error will only be caused by the higher-orderterms in the Lamb-Dicke expansion (57). For N =
2, we get ε mot ≈ × π N ( N − )( η x ) ( ¯ n x + ¯ n x ) / N , where we haveassumed a thermal state for the transverse vibrational modes with mean phonon number ¯ n x . In Fig. 3, we represent thefull error as a function of the gate time for different dephas-ing rates, choosing Ca + qubits to compare with the previousgates. As announced earlier, this figure shows that the error offast gates is dominated by the off-resonant carrier error. Wenote that the optimum transverse gates shown in this figureare faster and more accurate than the longitudinal and trans-verse gates of Figs. 1 and 2. Regarding the comparison to thelongitudinal-gate performance of Fig. 1, we note that the setof axial trap frequencies used in Fig. 2 is always below theaxial trap frequency of Fig. 1 (see the particular values in bothcaptions). Accordingly, the performance and speed shown inFig. 1 sets an upper bound for the comparison of axial andtransverse gates, and one concludes that the the transverseMS gate can indeed achieve higher speeds and fidelities. InSec. III C 2, we will discuss how to increase the gate speedeven further, while achieving also higher fidelities, in trapswhere the intrinsic radial micromotion can be exploited. (ii) Multi-pulse entangling gates: In the previous section,we have shown how to increase the speed of single-pulse MSgates by increasing the laser intensity. Let us now address analternative strategy to speed up the entangling gates by consid-ering a multi-pulse scheme with N p pulses (87). In addition toincreasing the laser intensity, one can also explore how to dis-tribute it among the different pulses in order to attain higherspeeds without compromising the gate fidelities.To analyse this multi-pulse scheme, we need to find the par-ticular expression for the time-evolution operator in Eqs. (81)and (83). By performing the corresponding integrals inEqs. (84) and (85), we find the following state-dependent dis-placements and phonon-mediated interaction strengths γ i , m ( t g ) = ∆ γ N p i , m , ∆ γ n p i , m = n p ∑ n (cid:48) p = γ n (cid:48) p i , m , g ( t g ) = − i J n p t g + δ g + ∑ m N p ∑ n p = ∆ γ n p − , m (cid:0) γ n p , m (cid:1) ∗ − H . c .. (104)Here, we have introduced the following constants γ n p i , m = f n p i , m x α m (cid:16) C τ n p δ − ω m , α e i ( δ − ω m , α ) t n p + C τ n p − δ − ω m , α e − i ( δ + ω m , α ) t n p (cid:17) , J n p = ∑ m f n p , m x α m f n p , m x α m ω m , α δ − ω m , α , δ g = ∑ m ∑ n p f n p , m x α m f n p , m x α m C τ n p − δ + ω m , α + (cid:16) C τ n p δ + ω m , α − C τ n p δ (cid:17) e i2 δ t n p − δ + ω m , α + C τ n p δ + ω m , α + (cid:16) C τ n p − δ + ω m , α − C τ n p − δ (cid:17) e − i2 δ t n p δ + ω m , α , (105)and used the circle function, C τω = ( − e i ωτ ) / ω [40]. Asa consistency check, note that for a single CW pulse N p = , t n p =
0, the terms f n p i , m x m C τ n p ω with ω ≈ ω α can be neglectedby a rotating-wave approximation for | f n p i , m x α m | (cid:28) ( δ + ω m ) , which follows from Eq. (72). Accordingly, one gets the sim-plified expressions in Eqs. (88)-(89), which were the startingpoint in the analysis of the previous section.To illustrate how the CW schemes can be modified to im-6prove the gate speed, we focus on schemes of equidistantlaser pulses of identical widths [65, 66]. In this case, onehas τ n p = τ : = t g / N p , and t n p = τ ( n p − ) in Eq. (87). Theconditions γ i , m ( t g ) = ∑ n p Re (cid:8) z m , n p (cid:9) Ω α i , n p = , ∑ n p Im (cid:8) z m , n p (cid:9) Ω α i , n p = , (106)where z m , n p = C τ n p δ − ω m , α e i ( δ − ω m , α ) t n p + C τ n p − δ − ω m , α e − i ( δ + ω m , α ) t n p ,and we denote the Rabi frequencies for each of the pulses as Ω α i , n p . Therefore, for N ions and thus N normal modes along aparticular trap axis, one has a system of 2 N linear equations,and a non-trivial solution of Eq. (106) can be found if we allowfor N p = N + { Ω α i , n p / Ω α i , } N p n p = . Sincewe want to study the conditions that allow for a speed-up withrespect to the single-pulse gates in Eqs. (95) or (103), we shallfix the detuning to the corresponding optimal value, eitherEq. (96) or Eq. (99) for single/two-mode schemes. Hence,the only remaining equation comes from the condition to gen-erate a maximally-entangled state g i j ( t g ) = − i π /
8. This willsuffice to fix Ω α i , for a particular gate time, such that we cantarget pulse sequences that yield faster gates. (a) Addressing a single vibrational mode: Let us first ad-dress how to increase the speed of the single-pulse gates basedon the longitudinal CoM mode (Fig. 1) by exploiting a train ofequidistant pulses. For the longitudinal modes, the large fre-quency gap of the CoM mode with respect to other vibrationalmodes allows us to reduce the number of required pulses to N p =
3. We follow the above method to find the optimal pulsesequence for a fixed detuning and a certain gate time. Startingfrom the gate time of the highest-fidelity MS gates (see thestars in Fig. 1), we lower the target gate time, and search forpulse sequences that close the CoM phase-space trajectory fora fixed detuning (96) that does no longer fulfill Eq. (91).In order to assess quantitatively if the performance of thepulsed MS gate is also optimal, i.e. highest fidelity, weuse again the error model underlying Fig. 1, as discussedin the previous section. However, for the error due to theoff-resonant carrier, we consider ε carr ≈ N ( Ω z ) / δ with ( Ω z ) = ∑ n p ( Ω zn p ) / N p , which takes into account the distribu-tion of the Rabi frequencies within the pulse train. The resultsare presented in Fig. 4, which shows that one can obtain an ad-ditional speed-up by using a pulse train with state-dependentforces that alternate their direction. Moreover, the interme-diate pulse is very weak, which allows one to reduce the re-quired average Rabi frequency with respect to the single-pulsegates, and leads to a lower gate infidelity. If the multi-pulsedgate speed is increased above this optimum point, the infi-delity rises quickly due to the contribution of the off-resonantcarrier. This is the main difference with the more-demandingschemes [35–39] for arbitrary-speed gates that are not basedon the resolved-sideband regime (57). (b) Addressing both vibrational modes: We now study howto increase the gate speed of the single-pulse gates based onboth transverse modes (see Fig. 3). In this case, the vibrationalfrequencies are closely spaced, and we need to close all phase-space trajectories using N p = N + − t g ( µ s ) e g / T = . s e d e ph e ca rr
010 40 18012 e C W e mot − n p n p W z n p / w z . . .eps t g ( µ s ) e g / e d e p h e ca rr e C W e mot n p n p T = . s .
50 250 . n p n p n p . . W z n p / w z W z n p / w z W z n p / w z Figure 4.
Multi-pulse MS gate with the axial CoM mode: (mid-dle panels) State infidelity ε g for a pulsed MS gate mediated by thelongitudinal mode of a N = Ca + ions. We assumean axial trap frequency ω z / π = . η z = . n z = . ε g for a N p = T = . s (left), and T = . s (right),whereas the dotted lines represent the contributions of dephasing ε deph , motional ε mot , and carrier ε carr errors, and the gate infidelity ofa single-pulse (CW) gate ε CW , as indicated in the captions. The yel-low stars represent the optimum single-pulse gate times correspond-ing to r = above method to find the optimal pulse sequence for a fixeddetuning and a certain gate time. Starting from the gate timeof the highest-fidelity MS gates (see the stars in Fig. 3), welower the target gate time, and search for pulse sequences thatclose all phase-space trajectories for a fixed detuning (99) thatdoes no longer fulfill Eq. (103).In Fig. 5, we represent the estimated infidelity of a N p = N =
20 40 60 80 100 12000.0010.0020.0030.0040.0050.0060.0070.0080.0090.01 − − t g ( µ s ) e g / T = . s e d e ph e ca rr
810 40 n p . n p . t g ( µ s ) e g / e d e p h e ca rr n p n p T = . s . . . e C W
120 15050 e C W n p n p . W x n p / w x W x n p / w x W x n p / w x W x n p / w x Figure 5.
Multi-pulse MS gate with both transverse modes: (middle panels) State infidelity ε g for a pulsed MS gate mediatedby both transverse modes of a N = Ca + ions. Weassume a radial trap frequency ω x / π = . η x = . n x = .
05 for the CoM mode. The yellowsolid lines correspond to the total state infidelity ε g for a N p = T = . s (left), and T = . s (right), whereas the dotted lines represent the contributions of de-phasing ε deph , motional ε mot , and carrier ε carr errors, and the gate in-fidelity of a single-pulse (CW) gate ε CW . The yellow stars representthe optimum single-pulse gate times corresponding to r = r = contribution of the off-resonant carrier.Although these results show that the error reduction bymoving onto pulsed MS gates is not that large, the improve-ment in gate speed with respect to the optimal single-pulsegate can be substantial if one only wants to maintain the gateerror to the same level. As discussed previously, increasingthe speed even further in both of these pulsed schemes leadsto an increase in the infidelity due to the off-resonant carrier.In the following section, we explore the advantage of exploit-ing the intrinsic micromotion to improve the gate speed evenfurther, while simultaneously maintaining error rates below agiven threshold.
2. Entangling gates with micromotion forces
After this long exposition, we have all the required ingre-dients to understand how the different MS gate schemes pre-sented above can be improved by exploiting the ion-crystalintrinsic micromotion. Considering the regime (78), one canuse directly the previous equations for the secular MS gatesdiscussed in Sec. III C 1, but taking into account the particu-lar expressions for the micromotion off-resonant carrier (77)and the micromotion state-dependent forces (80). This sim-ply amounts to substituting in all equations of Sec. III C 1: thelaser MS detunings by δ → ˜ δ , the Rabi frequencies of the sec-ular state-dependent forces by Ω α i → ˜ Ω α i = Ω α i q α /
4, and theerror due to the off-resonant carrier by ε carr ≈ N ( Ω α ) / δ → ˜ ε carr = N ( ˜ Ω α ) / q α Ω , where we have further assumed thatmicromotion compensation fulfils˜ β i (cid:28) q α ω α Ω rf , (107)which is consistent with the experimentally-achieved valuesthat will be discussed in Sec. IV A. This equation gives a prac-tical bound on how small the excess micromotion must be inorder for our analysis to be correct.From these substitutions, one observes that the strength ofthe micromotion state-dependent dipole forces is reduced withrespect to the one of the state-dependent secular forces (68)by a factor of q α /
4. Therefore, more powerful lasers willbe required to achieve the typical speed of secular MS gatesin Figs 1-5. However, provided that such laser sources areavailable, the maximum Rabi frequency will not be limited by | Ω α i | (cid:28) δ as occurred for the secular MS scheme (72), butinstead by | Ω α i | (cid:28) Ω rf . Hence, exploiting the intrinsic micro-motion, one can either maintain the gate speed while increas-ing the gate fidelity achieved by the secular MS schemes, orvice versa.Qualitatively, for the same gate speed, the leading carriererror for micromotion-enabled MS gates ˜ ε carr and secular MSgates ε carr is related by ˜ ε carr = ε carr ( δ / q α Ω rf ) . Hence, thecarrier error will be reduced provided that δ < q α Ω rf , (108)where δ ∼ ω α . As announced below Eq. (79), the advantageof the scheme will be larger, the smaller the ratio ω α / Ω rf canbe made in the experiment. The microscopic trap parameter q α /
4, which controls the relative amplitude of the intrinsicmicromotion and the secular oscillations (14), sets how smallis the ratio ω α / Ω rf required to be for the scheme to be advan-tageous. From a different perspective, this inequality showsthat the coupling to the first micromotion sideband has to besufficiently big for the scheme to become advantageous.Conversely, if we want to increase the gate speed but main-tain the fidelity of the secular MS gates, one can show that thegate times t g of single-pulse secular schemes in Eqs. (95) orEq. (103) are related to the micromotion-enabled gate times˜ t g as follows ˜ t g = t g ( δ / q α Ω rf ) . Accordingly, provided thatthe parameter regime (108) is achieved, there will be a speed-8 − − − − − − T = . s T = . s T = . s T = . s e g W rf / w z
200 800 100010 W rf / w z T ˜ t g ( µ s ) a T = . s T = . s T = . s T = . s e g T ˜ t g ( µ s ) b
20 80 100 W rf / w x W rf / w x Figure 6.
Micromotion-enabled improvement of single-pulse MS gates: (a) (main panel) State infidelity ˜ ε g for a single-pulse micromotionMS gate mediated by the longitudinal modes of a N = Ca + ions with ω z / π = . η z = . n z = . q z = .
03, and vary the r.f. frequency Ω rf . The dotted line corresponds to Ω rf = δ / q z ,and thus separates the region where the micromotion-enabled MS gates are advantageous (right) and disadvantageous (left shaded region). Werepresent the corresponding gate times ˜ t g in the inset. The circles in the main panel and inset coincide with the performance of the optimalsecular MS gates shown in Fig. 1 as yellow stars. Hence, the right region describes a micromotion-enabled improvement in both gate speedand fidelity. (b) Same as (a) but for a single-pulse micromotion-enabled MS gate mediated by both transverse modes of a N = Ca + ions with ω x / π = . η x = . n x = .
05 for the CoM mode. In this case, we consider a radial micromotion parameter q x = .
3. Two of the circles in the main panel and inset coincide with the performance of the optimal secular MS gates shown in Fig. 3 asyellow stars. The micromotion-enabled improvement of these transverse CW gates is qualitatively similar to the longitudinal ones in (a) . up of the entangling gates. A similar speed-up will also takeplace for the multi-pulsed MS gates.To be more quantitative, we now study the total gate in-fidelity ˜ ε g for the micromotion-enabled version of the secu-lar MS schemes of Sec. III C 1. Therefore, in addition to thechange in the carrier error already discussed, we also considerthe dephasing and motional contributions to the gate infidelity.We extract the optimal gate time ˜ t g that minimizes the gate in-fidelity ˜ ε ming , and represent these two quantities as a functionof the ratio Ω rf / ω α , which determines the region where themicromotion scheme becomes advantageous (108).In Fig. 6, we study the micromotion version of the single-pulse secular MS gates mediated by longitudinal (Fig. 1) andtransverse (Fig. 3) phonon modes. The circles correspond tor.f. frequencies that fulfill Ω rf = δ / q α , such that the perfor-mance of the micromotion-enabled gates coincides with thatof the standard secular MS gates. For larger r.f. frequencies(non-shaded regions), the micromotion scheme provides si-multaneously lower gate errors (main panel) and lower gatetimes (inset), both for the MS gates mediated by longitudinal(Fig. 6 (a) ) and transverse (Fig. 6 (b) ) vibrational bus modes.A similar improvement is found in Fig. 7 for the micromo-tion version of the multi-pulse secular MS gates mediated bylongitudinal (Fig. 4) and transverse (Fig. 5) phonon modes.Let us remark that this micromotion-enabled improvementof multi-pulse MS gates differs from the results presented inRef. [44]. Here, C. Shen et al. derive sequences for fast en-tangling gates to mitigate the adversarial effect of the excessmicromotion of planar crystals. In our scheme, we exploit theintrinsic micromotion instead, and turn its effect into a featurethat may allow one to improve on both fidelity and speed of phonon-mediated entangling quantum gates. IV. EXPERIMENTAL CONSIDERATIONS
In this section, we discuss the experimental prospects ofreaching the required parameter regime that would lead to themicromotion-enabled improvement of the entangling gatesdescribed above. We start by discussing in Sec. IV A the state-of-the-art excess micromotion compensation, and the possi-bility of reaching the desired range in Eq. (107). In Sec. IV B,we discuss the difficulty of fulfilling Eq. (108) with currenttrap designs, and the prospects of satisfying it with realistictrap designs that may become accessible in the future.
A. Compensation of excess micromotion
Excess micromotion can give rise to a series of undesiredeffects [49], such as (i) a parametric heating that can increasethe secular motion of ion crystals, limiting the temperaturesachieved by laser cooling or even inducing crystal instabili-ties [59], (ii) a laser heating for parameters where laser coolingwould be expected in the absence of excess micromotion [59],which can also be caused by the intrinsic micromotion [60],and (iii) motional shifts of frequency standards (e.g. second-order Doppler shifts) [49]. Therefore, a great deal of experi-mental effort has been devoted over the years to develop meth-ods for a precise estimation and minimization of the excessmicromotion. These methods range from (a) monitoring thechange of the ion equilibrium position as the secular trap fre-9 − − − − − − − T = . s T = . s T = . s T = . s e g W rf / w z
200 800 100010 W rf / w z T ˜ t g ( µ s ) a T = . s T = . s T = . s T = . s e g T ˜ t g ( µ s ) b
20 80 100 W rf / w x W rf / w x Figure 7.
Micromotion-enabled improvement of multi-pulse MS gates: (a) (main panel) State infidelity ˜ ε g for a multi-pulse micromotionMS gate mediated by the longitudinal modes of a N = Ca + ions with ω z / π = . η z = . n z = . q z = .
03, and vary the r.f. frequency Ω rf . The dotted line corresponds to Ω rf = δ / q z ,and thus separates the region where the micromotion-enabled MS gates are advantageous (right) and disadvantageous (left). We represent theassociated gate times ˜ t g in the inset. Two of the circles in the main panel and inset coincide with the performance of the optimal secular MSgates shown in Fig. 4 as orange stars. (b) Same as (a) but for a multi-pulse micromotion-enabled MS gate mediated by both transverse modesof a N = Ca + ions with ω x / π = . η x = . n x = .
05 for the CoM mode. In this case, we consider a radialmicromotion parameter q x = .
3. The circles in the main panel and inset coincide with the performance of the optimal secular MS gates shownin Fig. 5 as orange stars. quencies are modified, to (b) comparing the fluorescence in-tensities of emitted photons when the lasers are tuned eitherto the bare carrier δ ≈ δ ≈ Ω rf (i.e. resolved-sideband regime), and (c) monitoring cross cor-relations of the time delay between the emitted photons andthe r.f. signal (i.e unresolved-sideband regime). The precisionof method (a) is limited by the resolution limit of the opticsthat measures the ion position, whereas that of (b,c) dependson limitations and noise on the laser and r.f. sources.Provided that one of these methods yields an accurate mea-surement of excess micromotion, one can either apply addi-tional electric fields to compensate the force of the spuriousd.c. fields (5) due to patch potentials or unevenly coated elec-trodes, or load the electrodes with reactances to compensatethe spurious asymmetries leading to the oscillating force ofthe a.c. fields in Eq. (5) (see the discussion in [49]). A de-tailed account of the achieved minimization of excess micro-motion from different experimental groups can be found inRef. [52], which shows that a careful compensation with dif-ferent methods typically achieves β -parameters (64) on theorder of β i ∼ − . Using tightly-focused dipole beams toprobe the ion position can be exploited to achieve even bet-ter micromotion compensation [62], so it is reasonable toconsider that the β -parameter can attain values in the range β i ∼ − -10 − . We note that a realistic value for the idealPaul trap parameters in Eq. (4) yields q α ∼ . . α = { x , y } , such that the desired com-pensation regime in Eq. (107) can be achieved with state-of-the-art trapped-ion technology. For the axial direction, con-sidering short segmented linear traps, one may achieve ratiosof q z / q x ≈ − [67]. Considering the performance of the ax-ial micromotion-enabled entangling gates of Figs. 6 (a) and 7 (a) for q z = .
03, the smaller values of q z for these segmentedtraps would require a much higher ratio of Ω rf / ω z , as wellas a much higher laser power to achieve similar gate speeds.Accordingly, finding experimental trap designs that meet therequirements for a micromotion-enabled improvement basedon axial modes seems very challenging, and this motivates usto consider the radial micromotion gates below. B. Discussion of current and future trap designs
The suggested scheme requires a large ratio of the drivefrequency to the secular motional frequency in the radial di-rection Ω rf / ω x . Since the confinement properties of ion trapscan be accurately described by Mathieu equations that are in-dependent on the actual trap geometry [68], a study based ona geometry that is suitable for usual trapping parameters willalso suffice to explore the possibility of reaching the requiredparameters for a micromotion-enabled improvement of the en-tangling gate. Current traps for quantum information process-ing operate usually in the regime of Ω rf / ω x ≈ −
20 [69, 70].Experimentally, multi-qubit gate operations with Ω rf / ω x = Ca + [71, 72]. Thisratio, together with the rest of the parameters used in Fig. 6 (b) ,would already yield a benefit from the micromotion-enabledentangling gates. To be more precise, assuming a decoher-ence time of T = .
8s and the error model described above,the single-pulse MS gate based on secular radial forces wouldreach ε g = · − in a time t g = µ s, whereas the onebased on micromotion radial forces could attain ε g = · − in a time t g = µ s. Let us note that to gain full advantageof the protocol, one would need even higher ratios of Ω rf / ω x ,0which have not been achieved yet in experiments.While there is no fundamental reason that will prohibitreaching even higher drive frequencies, one needs to takepractical considerations into account. The dissipated powerinside the trap will increase since the amplitude of the r.f.drive voltage needs to be increased, leading thus to a higherpower dissipation in the trap itself, and also in the electricalconnections to the trap [70]. Managing the increased heat loadwill require complex thermal management techniques, espe-cially in the context of cryogenic systems. In this context, asmaller trap and connection capacitance is beneficial as it willfacilitate the design of the required circuitry to generate theradio frequency trapping fields [70]. C. Technical noise sources
Estimating the error budget accounting for additional tech-nical limitations can, for the proposed gate scheme, be per-formed analogously to other high-fidelity entangling gate op-erations, as detailed for instance in Ref. [33]. Regarding thedifferences for the micromotion-enabled gates, let us notethat, in case that the experimentally available laser power islimited, the gate duration would be increased by a factor of1 / √ q α , which follows from the different scaling of the dipoleforces in Eqs. (73) and (80). In general, this would makethe gate more susceptible to dephasing noise. Accordingly, iflaser power is the limiting factor, one should consider contin-uous [25, 26] or pulsed [73] dynamical decoupling techniquesto combat this noise.Another technical aspect that would differ from entanglinggates that do not make explicit use of micromotion is thegeneration of the bichromatic light fields. In the presentedgate, the frequency of the beat note must be on the order of2 Ω rf ≈ π
100 MHz, whereas for the standard gate the modu-lation frequency is on the order of ω α / π ≈ V. CONCLUSIONS AND OUTLOOK
In this work, we have developed a set of theoretical tools toanalyze the effects of excess and intrinsic micromotion in theschemes for high-fidelity quantum logic gates with trapped-ion qubits. We have shown that, in situations where the excessmicromotion is compensated to a high degree, it is possibleto exploit the intrinsic micromotion to improve on both thespeed and fidelity of current schemes for entangling gates. Wehave derived a set of conditions that identify the parameterregime where such an improvement can occur, and discussedthe possible challenges of reaching this regime consideringrealistic experimental conditions.Aside from the particular gate scheme, we have presentedfor the first time a detailed quantum-mechanical treatment ofintrinsic and excess micromotion in arbitrarily-large chains oftrapped ions. This has allowed us to develop a generic theoryfor the laser-ion interaction in the presence of micromotion,which might be useful for future trapped-ion studies in com-pletely different contexts.
Acknowledgements.–
We thank J. Home, J. Alonso, and T.Mehlst¨aubler for useful conversations, and H. Landa and A.Lemmer for their comments and the careful reading of themanuscript.The research is based upon work supported by the Of-fice of the Director of National Intelligence (ODNI), Intel-ligence Advanced Research Projects Activity (IARPA), viathe U.S. Army Research Office Grant No. W911NF-16-1-0070. The views and conclusions contained herein are thoseof the authors and should not be interpreted as necessarilyrepresenting the official policies or endorsements, either ex-pressed or implied, of the ODNI, IARPA, or the U.S. Govern-ment. The U.S. Government is authorized to reproduce anddistribute reprints for Governmental purposes notwithstand-ing any copyright annotation thereon. Any opinions, findings,and conclusions or recommendations expressed in this mate-rial are those of the author(s) and do not necessarily reflect theview of the U.S. Army Research Office.We also acknowledge support by U.S. A.R.O. throughGrant No. W911NF-14-1-010. A. B. acknowledges supportfrom Spanish MINECO Project FIS2015-70856-P, and CAMregional research consortium QUITEMAD+. P. S., T. M. andR. B. acknowledge support from the Austrian Science Fund(FWF), through the SFB FoQus (FWF Project No. F4002-N16) and the Institut f¨ur Quanteninformation GmbH. [1] M. A. Nielsen and I. L. Chuang,
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