intrinsic anharmonic effects on the phonon frequencies and...

PHYSICAL REVIEW A 90, 053405 (2014)

Intrinsic anharmonic effects on the phonon frequencies and effective spin-spin interactionsin a quantum simulator made from trapped ions in a linear Paul trap

M. McAneny and J. K. Freericks*

Department of Physics, Georgetown University, 37th and O Streets NW, Washington, DC 20057, USA(Received 20 June 2014; published 6 November 2014)

The Coulomb repulsion between ions in a linear Paul trap gives rise to anharmonic terms in the potentialenergy when expanded about the equilibrium positions. We examine the effect of these anharmonic terms on theaccuracy of a quantum simulator made from trapped ions. To be concrete, we consider a linear chain of Yb171+

ions stabilized close to the zigzag transition. We find that for typical experimental temperatures, frequencieschange by no more than a factor of 0.01% due to the anharmonic couplings. Furthermore, shifts in the effectivespin-spin interactions (driven by a spin-dependent optical dipole force) are also, in general, less than 0.01% fordetunings to the blue of the transverse center-of-mass frequency. However, detuning the spin interactions nearother frequencies can lead to non-negligible anharmonic contributions to the effective spin-spin interactions.We also examine an odd behavior exhibited by the harmonic spin-spin interactions for a range of intermediatedetunings, where nearest-neighbor spins with a larger spatial separation on the ion chain interact more stronglythan nearest neighbors with a smaller spatial separation.

DOI: 10.1103/PhysRevA.90.053405 PACS number(s): 37.10.Ty, 03.75.−b, 03.67.Ac, 03.67.Lx

I. INTRODUCTION

In 1982, Richard Feynman opened the field of quantumsimulation when he proposed that quantum simulators can beemployed in order to study the evolution and interactions ofcomplex quantum mechanical systems [1]. It is only recentlythat ion trap quantum simulators have demonstrated successin engineering model spin-systems in both one-dimensionaland two-dimensional lattices of trapped ions [2–13]. Startingwith the demonstration of the effective spin interactionbetween two ions [2], it was shown that larger numbersof ions interact with well-defined Ising spin exchange [3],which can show frustration [4,5], and can be scaled toapproximate the thermodynamic phase transition [6]. Penningtrap experiments [7,8], showed that the same concepts can beextended to hundreds of ions trapped in a rotating triangularlattice. The idea of stroboscopic quantum simulation has alsobeen shown [9]. Recently, Paul trap systems have been scaledup to 18 ions [10] and properties of dynamics and excitedstates have been examined via Lieb-Robinson-like studiesof correlation growth [11,12] and spectroscopy of excitedstates [13]. These ion-trap systems work well due to theirlong decoherence times, scalability, and ability to be preciselycontrolled experimentally.

As the precision of these experiments grows, one needs toexamine perturbations of these systems that carry them awayfrom the simplest ideal. In addition, as the system sizes grow, itbecomes increasingly difficult to fully cool the systems downto low temperatures as Raman sideband cooling becomes morecomplex and difficult to carry out. Hence, it is often only thephonon modes that are to be driven that are cooled below theDoppler limit; the phonons in other spatial directions are oftenleft at the Doppler limit, which can have them with tens tohundreds of quanta excited.

Anharmonic effects enter into an oscillator when the periodof the oscillation depends on its amplitude. In solid state

*[email protected]

physics, anharmonic effects are well known in causing latticesto (typically) expand as they are heated. Another way ofdescribing this behavior is that as anharmonic terms are consid-ered, they break the simple picture of free normal modes thatoscillate at their own independent frequencies into a coupledoscillator system that can have its periods change, that can haveresonantly enhanced dissipations, and that can excite quantain the coupled modes. It is impossible to completely removeanharmonic effects from an ion trap, even if the trappingpotentials can be made purely harmonic, because there is anintrinsic anharmonicity that arises due to the Coulomb inter-action between the ions. In this work, we investigate whethersuch anharmonic effects are likely to cause inaccuracies ina quantum simulation. We should emphasize that we are notexamining any other effects that might modify the operation ofthese simulators. Our goal is to determine at what level intrinsicanharmonic effects enter and whether they are small enoughto be neglected in the analysis of the quantum simulator.

Anharmonic effects have been considered previously forlinear Paul traps. Marquet et al. showed how one can determinethe coupling tensors that arise due to the anharmonic natureof the Coulomb interaction and how one can use thosecouplings to resonantly dissipate energy from one mode tothe other modes via optical-mixing-like effects [14]. Thistransfer of energy from one mode to another was investigatedexperimentally in a two-site chain [15]. The effects of anhar-monicities in either the potentials or the Coulomb interactionwere investigated to see how phonon frequencies shift dueto the occupancy of other phonon modes [16]. Anharmonicquantum effects on the zigzag transition were investigated witha renormalization-group approach based on path integrals [17].In this work, we focus on how the intrinsic anharmonicityaffects the phonon frequencies and how these, in turn, affectthe Ising spin exchange couplings. Hence, we do not treatanharmonic effects in the trapping potential, but only thosethat arise from the Coulomb interaction.

The remainder of this paper is organized as follows: InSec. II, we discuss the formalism for determining anharmonic

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http://dx.doi.org/10.1103/PhysRevA.90.053405

M. MCANENY AND J. K. FREERICKS PHYSICAL REVIEW A 90, 053405 (2014)

effects, numerical results follow in Sec. III, and we concludein Sec. IV. Details of the anharmonic coupling tensors appearin the Appendixes.

II. THEORETICAL FORMULATION

We consider a chain of ions in a linear Paul trap, which usesa combination of static and time-dependent fields in order totrap ions. The precise behavior of this system often includesmicromotion due to the time-dependent fields, but it is welldescribed by a static pseudopotential when the ion equilibriumpositions lie at the nulls of their potential energy surface.We provide our analysis under the assumption that the staticpseudopotential approach is accurate for describing the motionof the ions in the trap.

The potential energy describing such a system of N ionsincludes both a term describing the Coulomb interactionbetween each pair of ions and a term related to the springenergy of each ion along the z axis (which will be the axisof longitudinal alignment for the ions). The dimensionlesspotential is then given by

V = 12

∑α=x,y,z

N∑i=1

β2αx2iα +

1

2

N∑i,j = 1i �= j

1

|ri − rj | , (1)

where the potential has been scaled by mω2z l20 , and the dimen-

sionless ion coordinates ri = (xix,xiy,xiz) have been renormal-ized by a characteristic length l0 = [kZ2e2/(mω2z )]1/3, with kthe Coulomb coupling constant, Z the charge on the ion, e thecharge of an electron, and m the mass of the ion. In addition, wehave βα = ωα/ωz, where ωα is the trapping frequency in theαth direction. We consider the case with βx = βy � βz = 1,which gives rise to a one-dimensional chain for the ions, ifthe number of ions N lies below a critical value. The choiceβx = βy is for convenience only, as experimental systemsusually change these to be different from one another by afew percent. The degeneracies that arise from setting themequal do not affect our general results, and allow us to use onefewer parameter in the numerical calculations.

From Eq. (1), the equilibrium positions are readily foundnumerically by using nonlinear optimization routines to findwhere the potential has a minimum and the force vanishes [14].Then, by expanding the potential to fourth order in thecoordinates of the ions about their equilibrium positions, oneobtains the Hamiltonian written in the phonon creation andannihilation operator basis as follows:

H =3N∑ν=1

εν

(â†ν âν +

1

2

)

+3N∑

ν,ν ′ν ′′ = 1Bν,ν ′,ν ′′ (âν + â†ν)(âν ′ + â†ν ′ )(âν ′′ + â†ν ′′ )

+3N∑

ν,ν ′,ν ′′,ν ′′′ = 1Cν,ν ′,ν ′′,ν ′′′ (âν + â†ν)(âν ′ + â†ν ′ )

× (âν ′′ + â†ν ′′ )(âν ′′′ + â†ν ′′′ ), (2)

where the scaled normal-mode (phonon) energies satisfy εν =�ων/(mω2z l

20), and the explicit values for the cubic and quartic

coupling tensors B and C are given in the Appendixes. The νsubscript denotes the specific normal mode, which is indexedfrom 1 to 3N .

At this stage, it is appropriate to treat these higher orderterms as a perturbation to the harmonic Hamiltonian becausethey should correspond to small corrections to the potentialwhen the ions remain close to their equilibrium positions.The third-order term creates no first-order shift to the energyspectrum, as it contains an odd number of creation andannihilation operators. Therefore, a second-order perturbationexpansion is required for that term. The quartic term givesrise to a first-order shift in perturbation theory and hence thatterm is also included. Then, using time-independent Rayleigh-Schrödinger perturbation theory, the anharmonic shifts can becalculated to first order in C and second order in B, as has beendone in previous work [16]. This approach will calculate thechanges to fourth order in the expansion of the frequencies andeven include some of the sixth-order corrections. To get theremaining sixth-order corrections would require us to expandthe Coulomb interaction two more orders to determine thesixth-order coupling tensor and evaluate its shift on the energylevels to first order in perturbation theory. That calculation andall other higher order corrections are beyond the scope of thiswork. While we cannot rigorously show that this approachdetermines all of the anharmonic effects, it should be ableto accurately calculate anharmonic effects for the smallestdeviations from the harmonic model.

Before proceeding, it is relevant to note that the anharmonicfrequency shifts of the center-of-mass modes (both transverseand longitudinal) can be found to identically vanish throughfourth order. This was shown explicitly to third order [14]and can be immediately generalized to all orders, because thecenter-of-mass mode decouples from anharmonic correctionswhen the trap potential is purely harmonic (as we have chosenhere, since we assume that the pseudopotential that describesthe trap has no anharmonic terms) and the inter-ion forcessatisfy Newton’s third law [18,19] (as they do for the Coulombinteraction). Hence the center-of-mass frequency is fixed at thecorresponding trap frequency.

In general, the shifts in frequency (scaled by the transversecenter-of-mass frequency, ωc.m.) can then be written as afunction of the occupation number of each mode as

�ων(nν,{nν ′ })ωc.m.

= �E(nν + 1,{nν ′ }) − �E(nν,{nν ′ })εc.m.

, (3)

where c.m. denotes the center of mass, and {nν ′ } is the setof occupation numbers for all ν ′ �= ν. This is an intuitivedefinition for the anharmonic frequency shifts, since it is therelative energy shift caused by adding one more phonon to thesystem with harmonic frequency ων (when there are alreadynν phonons in that mode and the other modes are occupiedaccording to {nν ′ }) [16]. The �E terms are the shifts in theanharmonic energies from the harmonic result and, hence,vanish when there is no anharmonicity. Their explicit form isgiven in Appendix C.

The ions in the trap often have an internal hyperfinestructure which can be mapped onto Ising spin variables.For the Yb171+ ion, one usually takes the clock states as the

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INTRINSIC ANHARMONIC EFFECTS ON THE PHONON . . . PHYSICAL REVIEW A 90, 053405 (2014)

“spin-up” and “spin-down” states. This internal (spin) degreeof freedom can be coupled to the motional degrees of freedomby applying a spin-dependent optical dipole force. This isusually done by applying red and blue detuned laser beams(ω ± μ) on top of a carrier beam at ω. By considering theac Stark effect caused by these beams and factorizing theresulting evolution operator, one can realize a spin-dependentoptical dipole force on the ions [20,21]. In this way, themotional degrees of freedom of the system are coupled tothe spin degrees of freedom, generating the following IsingHamiltonian when one has harmonic phonons:

HIsing =∑i,j

Ji,j (t)σxi σ

xj , (4)

with exchange coefficients that depend on time. The time-independent piece of the spin-spin couplings is [21]

Ji,j = F2O

4m

N∑ν=1

bxνi bxνj

μ2 − ω2ν, (5)

where FO is the magnitude of the spin-dependent opticaldipole force and bανi is the αth spatial component of theith ion’s transverse phonon eigenvector corresponding to theων mode. In this summation, we only take the modes thatlie in the direction of the driving force, which is typically the(transverse) α = x direction.

It is difficult to extend this derivation to include thecubic and quartic phonon-mode coupling terms in the phononHamiltonian, because the operator factorization of the evo-lution operator becomes much more complicated (see, forexample, Ref. [22], which shows how to factorize the evo-lution operator and describes the problems that arise fromnoncommuting operators). Therefore, rather than calculatingthese complicated terms, it is assumed that these terms aresmall because the ions do not deviate far from their equilibriumpositions. The factorization of the laser-ion evolution operatorbegins with rewriting the evolution operator in the interactionrepresentation with respect to the phonon Hamiltonian (whichis time independent). The full evolution operator is justa product of the evolution operator of the phonon-onlyHamiltonian and of the interaction operator written in theHeisenberg representation with respect to the phonon-onlyHamiltonian. In the harmonic case, this corresponds to thecreation and annihilation operators for the different phononnormal modes becoming time dependent with a phase factorthat is exp[±iωνt] for the creation and annihilation operator.If we include the anharmonic phonon potential terms into thephonon-only Hamiltonian, then the Heisenberg representation(with respect to the phonon-only Hamiltonian) of the creationand annihilation operators is more complicated, because theanharmonic terms do not have simple commutation relationswith the creation and annihilation operators. Analyzing thisexactly becomes cumbersome. In condensed matter physics,one often invokes a so-called quasiharmonic approximationto treat such complexities approximately. The quasiharmonicapproximation takes the anharmonic system and replaces itwith an equivalent harmonic one, where the phonon frequen-cies are chosen to vary with the phonon occupancies in thesame way that the phonon frequencies are shifted due to theanharmonic terms. Hence, we use the traditional modification

of the creation and annihilation operators by a time-dependentphase factor, exp[±i(ων + �ων(nν,{nν ′ }))t], that incorporatesthe anharmonic shifts. We use this approach in approximatelytreating the anharmonic effects on the effective spin-spininteractions. The corrections to these expressions must arisefrom additional commutator effects and should be smallerthan the energy shifts that we include. Once this is done,the derivation of the spin-spin interactions is unchanged fromthe harmonic case, with the exception of using the shiftedfrequencies.

Note that the average occupation number of each phononmode can be shown to be n̄ν = [exp(�ων/kBT ) − 1]−1 whenthe ion chain is in thermal equilibrium at a temperature T .Then, instead of calculating the anharmonic frequency shifts(and thus the anharmonic spin-spin interactions) in terms ofthe occupation numbers, one can write the average occupationnumbers as a function of the temperature, and thus �ων =�ων(T ). Specifically, there are two temperature limits relevantto current experimental efforts. The first regime is the Dopplercooling limit for all modes, where the temperature reachedis of the order of a few hundred microkelvins. The secondtemperature regime is Doppler cooling plus sideband coolingon the transverse modes. For our purposes, the sideband cool-ing will lead to effectively zero occupation of the transversemodes, but the longitudinal modes remain at the Dopplerlimit temperature. Our assumption for the cooling is that theDoppler cooling, which is applied to all of the modes, occursfrom a broad atomic resonance that is much broader thanthe bandwidth of the phonons, so that all of the phonons arecooled to a temperature that is proportional to the width of theatomic resonance. The sideband cooling works with a separatenarrow atomic transition which allows one to remove all ofthe phonons in a particular phonon band (like the transversemodes). Given a fixed temperature for the different phononmodes, one can then determine their occupation number fromsimple Bose statistics, as described above.

III. NUMERICAL RESULTS

In this section, we present numerical examples to illustratehow anharmonic couplings affect the frequencies and spin-spin interactions between ions in the linear Paul trap. Theparameters we use reflect typical parameters used in currentexperimental efforts. The longitudinal trapping frequency isωz = 2π × 500 kHz, and the transverse trapping frequenciesare given by βx = βy = 10. Furthermore, we consider a trapwith 24 ions arranged in it, which, for the above parameters,is the maximum number of ions without a zigzag transition(unstable modes); i.e., the ion equilibrium positions lie in alinear chain. The trapped ions are Yb171+. Note that for theseparameters, the Doppler cooling limit is set by the transversecenter-of-mass frequency and is of the order of kBT ≈ �ωc.m..This implies the longitudinal modes should have of the orderof 1 to 10 quanta excited at the Doppler cooling limit.

A. Frequency shifts

First, we examine the effects of the anharmonicities onthe frequencies of the modes. Figure 1 shows the anharmonicshifts of the longitudinal and transverse frequencies both for

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FIG. 1. (Color online) Anharmonic frequency shifts as a functionof the temperature of the phonon modes measured relative to thetransverse center-of-mass phonon frequency ωc.m.. The center-of-mass mode is shown in black in all panels, while the transverse zigzagmode is shown in dark gray (red) in (a) and (c). All other modes areshown in light gray (cyan). (a) Frequency shifts of transverse modesin the Doppler cooling limit, where both transverse and longitudinalphonons are at temperature T . (b) Shifts of the longitudinal modesin the Doppler cooling limit. (c) Shifts of transverse modes withDoppler and sideband cooling (zero transverse phonon occupation,so transverse phonons are in the ground state, while longitudinal onesare at temperature T ). (d) Shifts of longitudinal modes with Dopplerand sideband cooling (zero transverse phonon occupation).

Doppler cooling only and for Doppler cooling with sidebandcooling of the transverse modes.

A few trends are immediately noticeable. First, the shiftsremain smaller than the order of 10−4ωc.m. for the caseof Doppler cooling only [Figs. 1(a) and 1(b)] and smallerthan the order of 10−5ωc.m. for the case with sidebandcooling also [Figs. 1(c) and 1(d)]. This suggests not onlythat the anharmonic frequency shifts are relatively small, butalso that sideband cooling can suppress these shifts anotherorder of magnitude. Furthermore, it is worth noting that, asanticipated analytically, the shifts for both the transverse andthe longitudinal center-of-mass modes are 0 (black lines inFig. 1). In an experiment with the center-of-mass frequency ofthe order of a megahertz, the anharmonic shift would be belowthe order of 100 Hz for Doppler cooling and below 10 Hz forDoppler-plus-sideband cooling. We expect effects of the orderof 100 Hz to be experimentally observable, but smaller shiftswill be difficult to see and are unlikely to affect other aspectsof the experiments. In addition, other effects on the experimentare likely to be more important than these intrinsic anharmonicfrequency shifts, when the shifts become so small.

The frequency shift curves are most nearly linear intemperature. This may be surprising considering that theexpression for �E has a quadratic dependency on theoccupation numbers of different modes (cf. Appendix C).However, when we take �E(nν + 1,{nν ′ }) − �E(nν,{nν ′ }),the quadratic dependencies cancel out, and since nν is roughlylinear with temperature except at extremely low temperatures(in this case, temperatures below the Doppler limit), we findthat the frequency shifts are linear with temperature.

FIG. 2. (Color online) (a) Proportional shifts in spin-spin in-teractions for detunings above the center-of-mass mode. (b) Thesame for detunings above the fifth lowest transverse frequencymode. Both plots are shown for δ = 10−1, δ = 10−2, δ = 10−3,δ = 10−4, δ = 10−5, and δ = 10−6. Symbols show the shifts for all(24 × 23/2 = 276) spin-spin interactions Jij . Solid horizontal linesshow the corresponding average shift.

Next, we consider the shifts in the effective static spin-spin interactions. Particularly, we discuss how the spin-spininteractions are affected when μ = ων(1 + δ), for which wesay that the spin-spin interactions are detuned by δ abovemode ν. Figure 2 plots the proportional change in the spin-spin interactions between the harmonic and the anharmonicHamiltonians for Doppler cooling only with a temperature thatsatisfies kBT = 10�ωz ≈ �ωc.m.. Figure 2(a) shows the shiftsin the spin-spin interactions for detunings above the transversecenter-of-mass mode, while Fig. 2(b) has detunings above thefifth lowest transverse frequency. We choose a wide range ofdetunings for completeness, even if some may be difficult toachieve in an actual experiment.

Clearly, Fig. 2(a) exhibits negligible shifts, especially fordetunings smaller than δ = 10−2. For the δ = 10−1 detuning(black curve), the spin interactions are shifted by as muchas 1%, although the average is more toward the order of0.01%. Furthermore, as the detuning decreases by an order ofmagnitude, so do the shifts. In this vein, the smallest detuningof δ = 10−6 (magenta curve) shifts the spin-spin interactionsby a factor of approximately 10−9. These shifts are obviouslynegligible.

On the other hand, Fig. 2(b), which shows detuning abovethe fifth lowest frequency mode, exhibits quite differentbehavior. First, the shifts are on average about 10% for thelargest detuning (black curve). Furthermore, for detunings assmall as 10−5, there are still shifts by a factor of 1%, althoughthe average shift is more toward 0.01% for these smallerdetunings. Then, in general, the anharmonic effects on thespin-spin interactions are not negligible for this case, even forthe small detunings.

This phenomenon can be attributed to the fact that �ωc.m. =0. Since the shift in the center-of-mass frequency is 0, thechanges in Ji,j that one might expect to see due to the(μ2 − ω2c.m.)−1 term are largely suppressed. However, due tononzero shifts in frequency for other modes, detuning aboveother modes makes the spin-spin interactions more sensitiveto anharmonic effects.

Figure 3 shows how the spin-spin interactions between thefirst ion and every other ion shift as a function of temperature

053405-4


FIG. 3. (Color online) Dependence of the spin-spin interactionsdetuned above the center-of-mass mode as a function of thesystem’s temperature. Left: Doppler cooling only. Right: Dopplerplus sideband cooling. Spin-spin interactions plotted are J1j , that is,spin interactions between the leftmost ion in the chain and every otherion, and we use three detunings, corresponding to the three colors.The most significant trend in this plot is that the shifts are nearly linearas a function of temperature. Furthermore, in (a) and (b), and even in(c) and (d), there are clear outliers that shift much more than the otherlines. As expected, these lines plot spin interactions between ions oneither side of the chain; the shifts are proportionally large because thespin-spin interactions between distant sites are quite small to beginwith, hence the effects of these shifts on the dynamics of the system,even though they appear to be large, are likely to be rather small.

for detunings above the center of mass. The spin-spin inter-actions appear roughly linear as a function of temperature.Furthermore, particularly for the larger detunings, there aresome outliers that shift significantly more than the others.These shifts are for the interactions between the first ion and theion that is farthest away. Since these interactions themselvesare small for larger detunings and far distances, even smallchanges in the spin interactions will change the interactionbetween these sites by a significant amount. In fact, even forthe 10−1 detuning, many of the interactions change by no morethan a factor of 10−7.

Finally, we found an interesting trend in the harmonicspin-spin interactions that warrants further discussion. Figure 4shows the harmonic spin-spin couplings Ji,j ’s as a function ofthe distance between the interacting spins (all 24 × 23/2 =276 spin-spin couplings are plotted). As a whole, the graph isin fact fairly typical: the smallest detunings produce the largest

FIG. 4. (Color online) Harmonic spin-spin interactions as a func-tion of the distance between interacting spins. Note that for in-termediate detuning, δ = 10−2 [(red) square] and δ = 10−3 [(blue)triangle] curves, and for small separations between spins, there canbe increasing spin interactions for increasing separation between theions. This suggests that ions towards either end of the chain interactmore strongly with their neighbors than those in the middle of thechain do, a surprising result since ions in the middle of the chainare closer together. For larger distances, spin-spin couplings becomeapproximate power laws, as expected.

spin interactions (which are approximately constant), while thelargest detunings cause smaller spin-spin interactions that falloff like J = r−3. However, if one looks closely at the curvescorresponding to intermediate detunings δ = 10−3 and δ =10−2 [(blue) triangle and (red) square curves, respectively],then one notices local regions at small distances (less thanthe characteristic length) for which the spin interactionsincrease as the distance between the interacting ions increases.Essentially, since nearest-neighbor ions on the inside of thechain are compressed more closely together than neighbors onthe outer edges of the chain, this suggests that ions toward theoutside of the chain interact more strongly with their neighborsthan the inner ions do with their neighbors. This effect isalso readily seen for smaller detunings, although the scale ofFig. 4 does not easily show this. Such spin-spin couplingscould potentially allow for interesting types of spin models tobe examined, since the couplings change character—initiallygrowing with distance and then decaying, and they also showthat it is not always true that the spin-spin couplings canbe described by a simple power-law behavior, as is oftenassumed. This behavior is due to the fact that the phononmodes with frequencies close to the center-of-mass mode(such as the tilt mode) have larger phonon displacementsfor the ions farthest from the center than phonon modesfarther from the center-of-mass phonon frequency, hence theions farther away make a larger contribution to the spin-spininteraction coming from the tilt mode. As the detuning isincreased from 0, the tilt mode and other phonon modes withsimilar properties enter the summation that determines thespin-spin interactions with a higher weight than phonon modesthat emphasize motion toward the center of the chain. Thisresult gives rise to the anomalous increase in the spin-spin

053405-5


interactions for nearest neighbors and for second nearestneighbors at intermediate detunings. When one is detunedsufficiently far from the center-of-mass mode, then all modescontribute equally, and the spin-spin couplings decay like theinverse third power of the distance between the ions. Hence,once the detuning is large enough, this anomalous behaviordisappears.

IV. CONCLUSION AND DISCUSSION

In this work, we have treated the anharmonic effects in thelinear Paul trap due to the Coulomb interaction to fourth orderin order to consider the effects of anharmonic couplings on thenormal-mode frequencies and spin-spin interactions betweentrapped ions. We find that the frequency shifts are small (of theorder of 10−4ωc.m.) when only Doppler cooling is utilized andanother order of magnitude smaller when sideband coolingis also implemented. Furthermore, we find that spin-spininteractions that are detuned above the center-of-mass modechange by no more than a factor of 10−4 except at thelargest detuning considered. This lack of significant changeis a consequence of the fact that �ωc.m. = 0. However, forspin interactions detuned near other modes, the anharmoniccouplings can have an appreciable effect, as large as 10%for the largest detuning considered and as large as 1% forthe smaller detunings. Finally, we find that the spins towardthe ends of the ion chain counterintuitively interact morestrongly with their neighbors than do the spins on the insideof the chain for a wide range of detunings to the blue of thetransverse center-of-mass mode. The mechanism behind thisphenomenon follows by analyzing the contributions of the dif-ferent phonon eigenmodes to the spin-spin couplings. Themodes closest to the center-of-mass mode have the largestrelative phonon displacements for ions farthest from the centerof the trap, compared to those that have phonon frequenciesfarther from the center-of-mass frequency, where the ionmotion of the normal modes is dominated by ions closer to thecenter of the trap, and this results in the anomalous behavior.

ACKNOWLEDGMENTS

We acknowledge John Bollinger for suggesting this prob-lem to us and for valuable discussions. We also thank CrystalSenko and Phil Richerme for valuable discussions. This workwas supported by the National Science Foundation under GrantNo. PHY-1314295. J.K.F. also acknowledges support from theMcDevitt bequest at Georgetown University.

APPENDIX A: DERIVATION OF THE ANHARMONICCOUPLING HAMILTONIAN

In order to generate the Hamiltonian given by Eq. (2),first we expand the potential to fourth order about theequilibrium positions of the ions, and then we transform thisresult from the ion-position basis to the phonon-mode basis.Let H0 denote the harmonic Hamiltonian for the system.Then

H = H0 + 16

∑α,β,γ

= x,y,z

N∑i,j,k = 1

∂3V

∂xαi∂xβj ∂xγ k

∣∣∣∣0

αiβj γ k

+ 124

∑α,β,γ,δ

= x,y,z

N∑i,j,k,l = 1

∂4V

∂xαi∂xβj ∂xγ k∂xδl

∣∣∣∣0

× αiβj γ kδl, (A1)where αi is given by αi = xαi − x0αi for equilibrium positionsx0αi and where the partial derivatives are evaluated at theequilibrium positions.

Now define

B̃αi,βj,γ k = ∂3V

∂xαi∂xβj ∂xγ k

∣∣∣∣0

(A2)

and

C̃αi,βj,γ k,δl = ∂4V

∂xαi∂xβj ∂xγ k∂xδl

∣∣∣∣0

. (A3)

These tensors can of course be solved for by taking the thirdand fourth derivatives of the potential [given by Eq. (1)]and then evaluating them at the equilibrium positions ofthe ions (which is what the subscript 0 indicates). However,the calculations themselves are quite tedious, so the lengthyalgebra is omitted. The final results for B̃ and C̃ are shown inAppendix B.

Now we must change to the phonon-mode basis. Inthe phonon-mode basis, the harmonic Hamiltonian is givenin terms of the creation and annihilation operators, andthe displacement from equilibrium αi is replaced by thephonon displacement operator Xν , which must be summedover all phonon modes and weighted by the normal-modeeigenvectors to yield the total displacement. The Hamiltonianbecomes

H = Hphon0 +1

6

∑{α,β,γ = x,y,z}

N∑{i,j,k = 1}

3N∑{ν,ν ′,ν ′′ = 1}

bανi bβν ′j b

γ ν ′′k B̃αi,βj,γ kXνXν ′Xν ′′

+ 124

∑{α,β,γ,δ = x,y,z}

N∑{i,j,k,l = 1}

3N∑{ν,ν ′,ν ′′,ν ′′′ = 1}

bανi bβν ′j b

γ ν ′′k b

δν ′′′l C̃αi,βj,γ k,δlXνXν ′Xν ′′Xν ′′′ , (A4)

053405-6


where the symbol bανi denotes the eigenvector of the harmonic phonon Hamiltonian for the νth mode, showing the displacementof the ith ion in the αth direction [14]. Define the ladder operators

âν =√

1

2εν

(ων

ωzXν + iPν

),

(A5)

â†ν =√

1

2εν

(ων

ωzXν − iPν

),

which satisfy the canonical commutation relations [aν,a†ν ′ ] = δνν ′ . Then, by expressing the Hamiltonian in terms of the phonon

creation and annihilation operators, we obtain [14]

H =3N∑ν=1

εν

(â†ν âν +

1

2

)+

3N∑ν,ν ′,ν ′′ = 1

Bν,ν ′,ν ′′ (âν + â†ν)(âν ′ + â†ν ′ )(âν ′′ + â†ν ′′ )

+3N∑

ν,ν ′,ν ′′,ν ′′′ = 1Cν,ν ′,ν ′′,ν ′′′ (âν + â†ν)(âν ′ + â†ν ′ )(âν ′′ + â†ν ′′ )(âν ′′′ + â†ν ′′′ ), (A6)

where

Bν,ν ′,ν ′′ = 16

(�

2ml20

)3/2(ωνων ′ων ′′ )

−1/23N∑

{i,j,k = 1}

∑{α,β,γ=x,y,z}

B̃αi,βj,γ kbανi b

βν ′j b

γ ν ′′k (A7)

and

Cν,ν ′,ν ′′,ν ′′′ = 124

(�

2ml20

)2(ωνων ′ων ′′ων ′′′ )

−1/23N∑

{i,j,k,l=1}

∑{α,β,γ,δ = x,y,z}

C̃αi,βj,γ k,δlbανi b

βν ′j b

γ ν ′′k b

δν ′′′l . (A8)

Note that B and C are simply the coefficients of the third- and fourth-order potential terms (B̃ and C̃ are in the ion-position basis)written in the phonon basis with some constants absorbed.

APPENDIX B: EXPRESSIONS FOR B̃ AND C̃

We now evaluate the exact expressions for the cubic and quartic phonon-mode coupling tensors in terms of the equilibriumpositions, as follows:

α = {x,y} : B̃αi,αj,zk =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−3 ∑Nh = 1h �= i

sgn(h−i)∣∣x0zh−x0zi∣∣4 , i = j = k;3 sgn(j−i)|x0zj −x0zi |4

, two indices are i, the other is j ;

0, i �= j,j �= k,k �= i.

B̃zi,zj,zk =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

6∑N

h = 1h �= i

sgn(h−i)∣∣x0zh−x0zi∣∣4 , i = j = k;−6 sgn(j−i)∣∣x0zj −x0zi∣∣4 , two indices are i, the other is j ;0, i �= j,j �= k,k �= i.

α = {x,y} : C̃αi,αj,αk,αl =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∑Nh = 1h �= i

9∣∣x0zh−x0zi∣∣5 , i = j = k = l;−9∣∣x0zi−x0zj ∣∣5 , three indices are i, the other is j ;9∣∣x0zi−x0zj ∣∣5 , two indices are i, the other two are j ;

0 otherwise.

α,β = {x,y}; α �= β : C̃αi,αj,βk,βl =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∑Nh = 1h �= i

3∣∣x0zh−x0zi∣∣5 , i = j = k = l;−3∣∣x0zi−x0zj ∣∣5 , three indices are i, the other is j ;3∣∣x0zi−x0zj ∣∣5 , two indices are i, the other two are j ;

0 otherwise.

053405-7


α = {x,y} : C̃αi,αj,zk,zl =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∑Nh = 1h �= i

−12∣∣x0zh−x0zi∣∣5 , i = j = k = l;12∣∣x0zi−x0zj ∣∣5 , three indices are i, the other is j ;

−12∣∣x0zi−x0zj ∣∣5 , two indices are i, the other two are j;0 otherwise.

C̃zi,zj,zk,zl =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∑Nh = 1h �= i

24∣∣x0zh−x0zi∣∣5 , i = j = k = l;−24∣∣x0zi−x0zj ∣∣5 , three indices are i, the other is j ;24∣∣x0zi−x0zj ∣∣5 , two indices are i; the other two are j ;

0 otherwise.

APPENDIX C: DERIVATION OF ANHARMONIC ENERGY SHIFTS VIANONDEGENERATE PERTURBATION THEORY

To solve for the energy shifts, we consider the first- and second-order Rayleigh-Schrödinger corrections for the quartic andcubic perturbations, respectively. The anharmonic energy shifts are then given by the following:

�E({nν}) = 0〈{nν}|V (4) |{nν}〉0 +∑

{mν }�={nν }

|0〈{mν}|V (3) |{nν}〉0 |2E0n − E0m

. (C1)

Recall that the first-order correction for the third-order potential term is 0 since there are an odd number of creation andannihilation operators. Furthermore, recall that we ignore the second-order correction for the fourth-order potential term.

Keeping in mind that E0n =∑3N

ν=1 εν(nν + 12 ) and that

V (3) =3N∑

ν,ν ′,ν ′′ = 1Bν,ν ′,ν ′′ (â

†ν + âν)(â†ν ′ + âν ′)(â†ν ′′ + âν ′′ ), (C2)

V (4) =3N∑

ν,ν ′,ν ′′,ν ′′′ = 1Cν,ν ′,ν ′′,ν ′′′ (â

†ν + âν)(â†ν ′ + âν ′ )(â†ν ′′ + âν ′′ )(â†ν ′′′ + âν ′′′ ), (C3)

one can directly solve for the anharmonic energy shifts. After some tedious algebra, the final expression for the anharmonicenergy shifts becomes

�E({nν}) = 33N∑ν=1

[(2n2ν + 2nν + 1

)Cν,ν,ν,ν + 2(2nν + 1)

3N∑ν ′ �=ν

(2nν ′ + 1)Cν,ν,ν ′,ν ′]

+(

ω2z l20m

�

)[−

3N∑ν=1

B2ν,ν,ν30n2ν + 30nν + 11

ων− 18

3N∑ν=1

3N∑ν ′ �=ν

Bν ′,ν ′,νBν,ν,ν(2nν ′ + 1)(2nν + 1)

ων

+ 93N∑ν=1

3N∑ν ′ �=ν

B2ν ′,ν ′,ν

(−4ων ′ (2nν ′ + 1)(2nν + 1)4ω2ν ′ − ω2ν

+ 2(n2ν ′ + nν ′ + 1

)4ω2ν ′ − ω2ν

− (2nν ′ + 1)2

ων

)− 18

3N∑ν=1

3N∑ν ′ �=ν

3N∑ν ′′ �= νν ′′ �= ν ′

Bν ′,ν ′,ν

×Bν ′′,ν ′′,ν (2nν′ + 1)(2nν ′′ + 1)

ων+ 36

3N∑ν=1

3N∑ν ′ �=ν

3N∑ν ′′ �= νν ′′ �= ν ′

B2ν,ν ′,ν ′′

(− (ων + ων ′)(1 + nν + nν ′ )(2nν ′′ + 1)

(ων + ων ′ )2 − ω2ν ′′

+ ων ′′(1 + nν + nν ′ + 2nνnν ′)(ων + ων ′)2 − ω2ν ′′

+ (ων − ων ′)(nν − nν ′)(2nν ′′ + 1)(ων − ων ′)2 − ω2ν ′′

+ ων ′′(nν + nν ′ + 2nνnν ′)(ων − ων ′ )2 − ω2ν ′′

)]. (C4)

This formula is used to evaluate the anharmonic shifts in the phonon frequencies throughout this paper. Note that these formulasare at most quadratic in the phonon occupation numbers and become linear when differences are taken with neighboringoccupancies. Hence, we only need the average phonon number in each mode for the thermal distributions to determine thetemperature-dependent shifts.

053405-8


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