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APS/123-QED Pattern formation with trapped ions Tony E. Lee and M. C. Cross Department of Physics, California Institute of Technology, Pasadena, California 91125, USA (Dated: January 28, 2011) Ion traps are a versatile tool to study nonequilibrium statistical physics,dueto the tunability of dissipationandnonlinearity. Weproposeanexperimentwithachainoftrappedions,wheredissipa- tion is provided by laser heating and cooling, while nonlinearity is provided by trap anharmonicity and beam shaping. The collective dynamics are governed by an equation similar to the complex Ginzburg-Landau equation, except that the reactive nature of the coupling leads to qualitatively different behavior. The system has the unusual feature of being both oscillatory and excitable at the same time. We account for noise from spontaneous emission and find that the patterns are observable for realistic experimental parameters. Our scheme also allows controllable experiments 1 with noise and quencheddisorder. 1 0 PACSnumbers: 2 n Pattern formation is the emergence of structure in a a J nonlinear medium far from equilibrium [1, 2]. This phe- 6 nomenonoccursinmanysettings,includingfluids,chem- 2 ical reactions, plasmas, and biological tissues. In tra- ditional pattern-forming systems, the collective behav- ] ior is set by the material properties, and theoretical de- S P scriptions are often phenomenological. For example, in n. theBelousov-Zhabotinskyreaction,theconcentrationsof FIG. 1: Chain of ions, each damped by a red-detuned beam i chemicalreagentsoscillateintime andproducetraveling andexcitedbyablue-detunedbeam. Thebluebeamisalong nl waves. Itisacomplicatedreactioninvolvingmanyinter- the ion chain, while the red beams are at an angle. The [ mediate states and rate constants. Hence, it is difficult intensity of the red beams is lowest at the trap center. Each to experimentally control the behavior of the system. ion is in its own anharmonic DC potential well (not shown). 2 v On the other hand, ion traps allow an unprecedented 9 level of control using optical and electrostatic forces and order to create many trapping regions [10], and thereby 6 have led to impressive experiments in quantum comput- 5 make a chain of ions, each in its own potential well. By ing [3] and quantum simulation [4]. In this paper, we 3 changingthe DC voltages,one cantune the shape of the show how ions are also useful for studying pattern for- 1. potential for each ion and thus control the nonlinearity mation. Theadvantageofusingionshereistheabilityto 1 [11, 12]. Let x be the axial displacement of an ion from tune dissipationand nonlinearity in situ, thereby having 0 its trap center, d the distance between trap centers, ω 1 moreexperimentalcontrolandbeingabletoseedifferent o theharmonictrapfrequency,andα thecoefficientofthe : effects within the same system. o v anharmonic quartic term in the trap potential. Xi Weshowhowpatternsariseinachainofionsdrivenfar We apply near-resonant laser beams to heat and cool from equilibrium. The collective dynamics are governed each ion (Fig. 1). When the laser frequency is above r a by an equation similar to the complex Ginzburg-Landau resonance (blue-detuned), the ion feels an anti-damping equation, which is one of the most studied equations in forceduetotheDopplereffect. Whenthelaserfrequency physics. However, the presence of only reactive coupling isbelowresonance(red-detuned),theionfeelsadamping in the ion chainleads to novelbehavior: while othersys- force [13]. We apply a blue-detuned beam in the axial tems areeitheroscillatoryorexcitable,the ionchaincan directionalongtheionchain. Foreachion,wealsoapply be both at the same time. Our scheme is also useful a red-detuned beam at an angle φ with respect to the for studying synchronization and Anderson localization. trap axis. The red beam is shaped so that the intensity Ourworkismotivatedbyrecentexperimentsonthenon- is lowest at the trap center. The ion is then heated near lineardynamicsofsingleions: aphononlaser[5,6]anda the center and increasingly cooledaway from the center, Duffing oscillator [7]. We note that the model described so the ion oscillates with a large amplitude, determined in this paper is also applicable to an array of nanome- bythe balanceofheatingandcooling. Thedissipationis chanical resonators [8]. easily tuned by changing the beam intensity. First we describe the proposed experimental setup. A Weuseasingly-ionized,two-levelatomofmassmwith linear Paul trap uses an RF electric field for radial con- adipoletransitionofwavelengthλ=2π/kandlinewidth finement and a DC field for axial confinement [9]. We γ. Eachredandbluebeamhasdetuning ∆ω and+∆ω − use a segmented trap, which has many DC electrodes in and intensity I and I , respectively. Let I be the sat- R B s 2 uration intensity. All beams have the same polarization. 2kee2 3αoℓ2 8~k2γ3∆ωIB/Is b= , c= , ν = ,(6) We use low laser intensities so that the ion is unsatu- νmd3ω2 νω2 mω (γ2+4∆ω2)2 o o o rated. We assume the radial motion is cooled near the where t¯= µt/2 is rescaled time, b is the coupling, and c Doppler limit (due to the projection of the red beams), relates how an ion’s amplitude affects its frequency. We so that the Doppler shift is due mainly to the axial mo- stress that b and c are directly related to experimental tion. For mathematical convenience, we use counter- settings. In the absence of coupling, each ion oscillates propagating beams (but in practice one would use sin- with amplitude A =1, which corresponds to an ampli- gle beams). Then the total optical force on an ion is | | tude of 2ℓ in x. The noise functions are due to sponta- calculated rigorously to be [5], neousemissionandrepresentscatteringbytheredbeams 8~k2γ3∆ω I I (ηR,σR) and blue beams (ηB,σB), F = R cos2φ+ B x˙ , (1) (γ2+4∆ω2)2 (cid:18)−I I (cid:19) s s 1 Hδ(t¯ t¯)δ under the additional assumptions k x¨ γ2/4 and hηmR(t¯)ηnR(t¯′)i = 3hσmR(t¯)σnR(t¯′)i= co−s2′φ mn(7) | | ≪ k x˙ ∆ω. To get position-dependent damping, we ηB(t¯)ηB(t¯) = σB(t¯)σB(t¯) =Hδ(t¯ t¯)δ (8) | | ≪ h m n ′ i h m n ′ i − ′ mn choose to vary IR quadratically along the trap axis, ~(γ2+4∆ω2) H = , (9) 2 96mω2ℓ2∆ω x o I (x) = I , (2) R B (cid:18)ℓcosφ(cid:19) where H is a dimensionless measure of the noise. We now examine the spatiotemporal properties of whereℓisthecharacteristiclengthoftheintensitygradi- Eq. (5), ignoring the effect of noise for now. First note ent. Theintensityprofiledoesnotneedtotakethisform thatthe equationissymmetricunderthe transformation or even be symmetric. In fact, a different profile may b,c,A b, c,A . In the continuum limit, we let lead to interesting higher-order terms in Eq. (5) [14]. n → − − ∗n A A(X) and The ions are coupled through Coulomb repulsion. If n → the displacements are small relative to the inter-ion dis- dA d2A tance(ℓ d),theinteractionislinear,andthefree-space = A+ib (1+ic)A2A. (10) ≪ dt¯ dX2 − | | couplingdecreaseswiththecubeofthedistance. Numer- ically, we find that interactions farther than the nearest ThisissimilartothecomplexGinzburg-Landauequation neighbor do not affect the overalldynamics much, so we (CGLE) [15], except that the coefficient of d2A/dX2 is assume only nearest-neighbor interactions. The equa- purely imaginary. This is because the Coulomb force is tions of motion are, reactive,while the CGLEincludes bothreactiveanddis- sipativeinteractions. Thisgreatlymodifies thebehavior, 0 = d2 x +ω2x +α x3 µ 1 xn 2 dx as seen below. dt2 n o n o n− (cid:20) −(cid:16) ℓ (cid:17) (cid:21)dt n AccordingtoEq.(10),aplanewavesolutionA(X,t¯)= 2kee2 Fei(QX−ωt¯) satisfiesF =1andω =bQ2+c. Bylineariz- + [(x x )+(x x )]+χ (t),(3) md3 n− n−1 n− n+1 n ing around this solution [2], we find that the condition forstabilityisbc 0,andaslongasthisisfulfilled,there n=1,...,N,where k is the Coulombconstant, e is the ≥ e isnorestrictiononthewavenumberQ. SinceEq.(10)is proton charge, µ is the damping coefficient, inthecontinuumlimit,weexpectlongwavelengthwaves 8~k2γ3∆ωIB/Is (wavelength at least severalions) when bc≥0. µ = , (4) When bc 0, it turns out that Eq. (5) allows short m(γ2+4∆ω2)2 ≤ wavelengthwavesthatarenotcapturedinthecontinuum and χn(t) is the noise. In this scheme, the inherent limit. DefineAñ =−A∗n forevennandAñ =A∗n forodd source of noise is spontaneous emission, since each emis- n and consider the continuum limit of the transformed sion causes a momentum kick ~k in a random direction. system. A plane wave A˜(X)=F˜ei(Q˜X−ω˜t¯) satisfies F˜ = We work in the regime where the nonlinearities and 1 and ω˜ = b(Q˜2 4) c, and the stability condition for − − interactionsaresmallperturbationstotheharmonicmo- long-wavelengthwavesisbc 0. Alongwavelengthwave tion. We write x (t) = 2ℓRe[A (t)e iωt], so that the in A˜ corresponds to a very s≤hort wavelength wave in A. n n − complex amplitude A encodes the slowly varying am- Therefore,Eq.(5)hasstableplanewavesforallvalues n plitude and phase of the underlying harmonic oscilla- of b and c: long wavelength for bc 0 and short wave- ≥ tions. In the Supplemental Material, we find the am- length for bc 0. This behavior is different from the ≤ plitude equation, CGLE, which has stable plane waves for bc > 1 and is − otherwise chaotic [15]. Another difference is that in the dAn = A +ib( 2A +A +A ) (1+ic)A 2A CGLE, only a band of Q is stable. dt¯ n − n n−1 n+1 − | n| n However, boundary conditions affect the selection of +[ηR(t¯)+iσR(t¯)]A +[ηB(t¯)+iσB(t¯)], (5) plane waves. With periodic boundary conditions, the n n n n n 3 (a) (a) (b) A 80 π 1 A12 1.5 0.5 90 1 ∆θπ2 Re A 0 ¯t (b) −0.5 e m 0.5 0 −1 caled ti8900 0 −2 −1 c0 1 b = 21 38 0 resca3le90d time t¯ 400 s e −0.5 r (c) FIG. 3: System of two oscillators with b=1. (a) Bifurcation −1 diagramascvaries. Therearesupercriticalpitchforkbifurca- 87 tions at (c,∆θ) = (−1,0) and (1,π). There are supercritical 90 −1.5 Hopf bifurcations at (1.22,2.82), (1.22,3.47), (−1.22,−0.32), and (−1.22,0.32), which give rise to stable limit cycles (not 10 20 30 40 50 shown). Solid and dashed lines denote stable and unstable oscillator index fixedpoints, respectively. (b) A limit cycle at c=1.4. FIG.2: Spacetimeplotofchainof50ionswithnearestneigh- bor interactions and open boundaries. Re A is plotted using nomial of r2, 1 the color scale in the side bar. A is the complex amplitude of the underlying harmonic oscillations. (a) b =1 and c=1 0 = (c2−1)2r18−(c2−1)(c2−3)r16−(c2−3)r14 showing uniform phase synchrony. (b) Same, but with the (b2+1)(c2+1)r2+b2(b2+1), (14) expected noise from spontaneous emission. (c) b = 1 and − 1 c=−1showing anti-phase structure. which may be solved numerically. The bifurcation di- agram is quite rich: as b and c change, saddle-node, pitchfork, and Hopf bifurcations appear, disappear, and wave number (Q or Q˜) is a multiple of 2π/N. Open change criticality. An example is shown in Fig. 3a. For boundary conditions are simpler to implement and are some values of b and c, there are supercritical Hopf bi- equivalent to setting dA = 0 at the boundaries. Thus furcationsto stable limit cycles,inwhichthe amplitudes dX when bc > 0, the only allowed plane wave is the Q = 0 and relative phase oscillate (Fig. 3b). The system is at wave,in which the ions are uniformly in-phase (Fig. 2a). least bistable for c < b, but certain values of b and c (This also occurs in the Belousov-Zhabotinsky reaction have four stable fi|xe|d p|oi|nts. in the absence of spatial inhomogeneities [2]). One can As N increases, there are still in-phase (∆θ = 0) and induce Q = 0 waves by, for example, changing the trap anti-phase (∆θ π) fixed points, although the region of frequency6ωo of one ion. The bc < 0 case is different, multistability in≈bc space shrinks. For c . b, there are becauseopenboundaryconditionsmeansettingA˜=0at alsomultiplestablelimitcycles,inwhic|h|the|e|ntirechain the boundaries. The final state is not a pure plane wave has the same average frequency or the chain is divided but a more complicated structure, in which the ions are into regions of different frequencies. almost uniformly anti-phase (Fig. 2c). A large ion chain is excitable in a novel way. An ex- We now examine the dynamics of two coupled ions, citable medium has the property that the uniform state expanding on previous work [16, 17]. First write Eq. (5) is stable to weak perturbations, but a perturbation that intermsoftherealamplitudesr1,r2andphasedifference exceedsathresholdgrowsrapidlyandthendecays. Usu- ∆θ =θ2 θ1, where An =rne−iθn, ally,amediumiseitheroscillatoryorexcitable[2]. How- − ever, the ion chain is both oscillatory and excitable at d∆θ b the same time. Suppose that bc > 0 and the chain is in = (r2 r2) c+ cos∆θ (11) dt¯ 2 − 1 (cid:18) r r (cid:19) the Q = 0 state (in-phase), with N large. We perturb 1 2 dr1 = (1 r2)r +br sin∆θ (12) the end ion A1 by δA. The Q = 0 state is linearly sta- dt¯ − 1 1 2 ble so a small perturbation decays away. But if δA is dr greaterthana threshold,itgeneratesa localizedpulseof 2 = (1 r2)r br sin∆θ . (13) dt¯ − 2 2− 1 anti-phase oscillations (Fig. 4). The pulse travels across the system, bouncing off the boundaries until it decays. Thissystemissymmetricunderthefollowingtransforma- Thistypeofexcitabilitydiffersfromtraditionalexamples tions: r ,r ,∆θ r ,r , ∆θ , c,∆θ c,π ∆θ , (likeneuronsandhearttissue),becausethepulseismade 1 2 2 1 { → − } { →− − } and b,∆θ b,π +∆θ . There are fixed points at of an alternating phase structure instead of an increased { → − } (∆θ,r ,r ) = (0,1,1) and (π,1,1), corresponding to in- chemical concentration. 1 2 phase and anti-phase motion. There is another set of Theexcitabilitycanbeintuitivelyunderstoodfromthe fixed points that correspond to roots of a quartic poly- fact that when bc > 0 and c . b, both in-phase and | | | | 4 fluctuate not only in time but in space as well. In our 0 0 scheme, the noise for each ion is independent and may be easily tuned by changingthe detuning ∆ω in Eq. (9). Onecouldobserve,forexample,noise-inducedtransitions 50 5 between stable fixed points in a small chain. ¯t e Another interesting use of the tunability is to study m d ti the effect ofquencheddisorder. A previousworkstudied e100 10 the mean-field version of Eq. (5) with random harmonic al esc frequenciesωo andfoundthatasbandcchange,thesys- r temundergoescontinuousanddiscontinuousphasetran- 150 15 sitions between the unsynchronized and synchronized states [8, 19]. It wouldbe interesting to study the lower- dimensional versions. We note that synchronization of 200 20 disparate oscillators is an important topic throughout 25 50 5 10 15 20 science [20, 21]. Furthermore,when the variance of ω is o oscillator index small, Eq. (5) may be mapped via the Cole-Hopf trans- formationto the Schrödinger equationfor a particle in a FIG. 4: Spacetime plot of Re A for chain of 50 ions showing anexcitationpulseforb=1andc=0.2. Perturbingthefirst random potential [22, 23]. The resulting pattern forma- oscillator beyond a threshold generates a pulse of anti-phase tion reflects the phenomenon of Anderson localization. oscillation. The initial conditions were A1 =−1 and An =1 We thank H. Ha¨ffner forhelpful comments. This work fortherest. Therightpanelisazoomed-inview. Colorscale wassupported by Boeing and NSF grantDMR-1003337. is the same as in Fig. 2. anti-phase oscillations may be stable for small chains, while only in-phase oscillation is stable for large chains. [1] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, Thus, a local region within a large chain may be anti- 851 (1993). phase for a short amount of time. A mathematical de- [2] M. C. Cross and H. Greenside, Pattern Formation and scriptionofthisphenomenonisleftopenforfuturework. Dynamics in Nonequilibrium Systems (Cambridge Uni- An important question is whether the patterns de- versity Press, Cambridge, 2009). scribed above would be observable for realistic experi- [3] D. Leibfried et al., Rev.Mod. Phys.75, 281 (2003). [4] K. Kim et al., Nature 465, 590 (2010). mental settings that satisfy all the theoretical assump- [5] K. Vahala et al., NaturePhysics 5, 682 (2009). tions. Indeed,wefindthatthe patternsarevisibleabove [6] S. Knu¨nzet al., Phys. Rev.Lett. 105, 013004 (2010). the noise from spontaneous emission. For example, the [7] N. Akerman et al., Phys. Rev.A 82, 061402(R) (2010). ion 24Mg+ has an S1/2 P3/2 dipole transition of wave- [8] M. C. Cross et al., Phys.Rev. Lett.93, 224101 (2004). − length λ = 279.6 nm and linewidth γ/2π = 42 MHz. [9] P. K. Gosh, Ion Traps (Clarendon, Oxford, 1995). Letting ω /2π = 100 kHz, I /I = 0.05, ∆ω = 6γ, [10] D. Kielpinski et al., Nature417, 709 (2002). o B s φ = π/4, ℓ = 30 µm, α /4π2 = 1015 Hz2/m2, and [11] W. K. Hensinger et al., Appl. Phys. Lett. 88, 034101 o d=500 µm,onefindsb=1.0,c=1.1,andH =5 10 4. (2006). × − [12] G.-D. Lin et al., Europhys. Lett. 86, 60004 (2009). Figure 2b shows that the Q = 0 state is clearly vis- [13] H. J. Metcalf and P. vander Straten,Laser Cooling and ible above the noise. Also, since many experiments Trapping (Springer,New York,1999). are already using large-scaletraps for quantum informa- [14] W.vanSaarloosandP.C.Hohenberg,PhysicaD56,303 tion [10], it should be straightforward to implement our (1992). scheme with many ions. [15] I. S. Aranson and L. Kramer, Rev. Mod. Phys. 74, 99 (2002). Intheexperiment,onecanmeasuretheamplitudeand [16] A. Kuznetsov et al., Physica D 238, 1203 (2009). phaseofA byrecordingwhenandwheretheionsscatter n [17] M.Grau,Seniorthesis,CaliforniaInstituteofTechnology photons [6, 7]. Changes in A occur on a time scale n (2009). t 1/µ,whichis muchslowerthan thatofthe harmonic [18] F. Sagués et al., Rev.Mod. Phys.79, 829 (2007). ∼ oscillation ( 1/ωo). Thus, one can observe dynamical [19] M. C. Cross et al., Phys.Rev. E73, 036205 (2006). ∼ effects, such as limit cycles and excitation pulses. The [20] A. S. Pikovsky, M. Rosenblum, and J. Kurths, Syn- entire bc space can be explored by, for example, tuning chronization: A Universal Concept in Nonlinear Science (Cambridge University Press, NewYork,2001). the parameters I and α . B o [21] J. A.Acebrón et al., Rev.Mod. Phys.77, 137 (2005). Itwouldbeinterestingtoturnupthenoisetoseewhat [22] H.Sakaguchiet al., Prog. Theor. Phys.79, 1069 (1988). happens. Itis knownthataddingnoise toa spatiallyex- [23] B. Blasius and R. Tönjes, Phys. Rev. Lett. 95, 084101 tendedsystemmayhavenontrivialeffects[18]. However, (2005). it is usually experimentally difficult to make the noise [24] S. Stenholm, Rev.Mod. Phys. 58, 699 (1986). 5 [25] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (Wiley, New York,1992). Supplemental Material This appendix provides supporting calculations for the main text. In the first section, we derive the amplitude equation using perturbation theory. In the second section, we calculate the noise from spontaneous emission. Derivation of amplitude equation Here we use perturbation theory to calculate how the weak nonlinearities and interactions in Eq. (3) of the main textchangetheamplitudeandphaseoftheharmonicoscillationsonalongtimescale. Weusethemethodofaveraging with amplitude and phase variables because the noise from spontaneous emission depends on the amplitude. First we rescale Eq. (3) of the main text with τ =ω t and y =x /ℓ, o n n d2 d 0 = y +y +αy3 ν(1 y2) y +D[(y y )+(y y )]+ζ (τ), (15) dτ2 n n n− − n dτ n n− n−1 n− n+1 n where α=α ℓ2/ω2, ν =µ/ω , D =2k e2/md3ω2, and ζ (τ) is the noise. Equation (15) describes a chain of van der o o o e o n Pol-Duffing oscillators. Let y = r (τ)cos[τ +θ (τ)], where r and θ change slowly (r˙ r , r¨ r˙ , θ˙ 1, θ¨ θ˙). Substituting this n n′ n n′ n ′ ≪ ′ ′ ≪ ′ ≪ ≪ into Eq. (15) and keeping the leading terms, 0 = 2r˙ sin(τ +θ ) 2r θ˙ cos(τ +θ )+αr3cos3(τ +θ )+ν[1 r2cos2(τ +θ )]r sin(τ +θ ) − n′ n − n′ n n n′ n − n′ n n′ n +D[2rn′ cos(τ +θn)−rn′−1cos(τ +θn−1)−rn′+1cos(τ +θn+1)]+ζn(τ). (16) Then multiply each equation by sin(τ +θ ) and integrate over the time interval [τ,τ +∆τ], where ∆τ is a multiple n of 2π, dr ν r2 D n′ = 1 n′ r + [r sin(θ θ )+r sin(θ θ )]+ξr(τ). (17) dτ 2 (cid:18) − 4 (cid:19) n′ 2 n′−1 n−1− n n′+1 n+1− n n Then multiply each equation by cos(τ +θ ) and integrate similarly, n dθ 3 D r r dτn = 8αrn′2+ 2 (cid:20)2− n′r−n′ 1 cos(θn−θn−1)− n′r+n′ 1 cos(θn−θn+1)(cid:21)+ξnθ(τ). (18) These equations describe how r and θ evolve. The noise functions are, n′ n 1 τ+∆τ ξnr(τ) = ∆τ Z dτ′ζn(τ′)sin(τ′+θn) (19) τ 1 τ+∆τ ξθ(τ) = dτ ζ (τ )cos(τ +θ ). (20) n r ∆τ Z ′ n ′ ′ n n′ τ To put Eqs. (17) and (18) in simpler form, rescale time (t¯=ντ/2) and amplitude (r =r /2), n n′ dr n = (1 r2)r +b[r sin(θ θ )+r sin(θ θ )]+ψr(t¯) (21) dt¯ − n n n−1 n−1− n n+1 n+1− n n dθ r r dt¯n = crn2 +b(cid:20)2− nr−n1 cos(θn−θn−1)− nr+n1 cos(θn−θn+1)(cid:21)+ψnθ(t¯). (22) where b = D/ν, c = 3α/ν, and ψr and ψθ are the rescaled noise functions. Then write everything in terms of a n n complex amplitude A = r e iθn, so that y (τ) = 2Re[A (τ)e iτ]. (We use e iθn instead of eiθn in order to match n n − n n − − up with the sign convention in Ref. [15].) Then A (t¯) evolves according to, n dA n = A +ib( 2A +A +A ) (1+ic)A 2A +ψA(t¯,A ), (23) dt¯ n − n n−1 n+1 − | n| n n n where ψA is the complex-valued noise function. n 6 Noise from spontaneous emission Here we calculate the expected noise from spontaneous emission. When an ion absorbs a photon from a laser, it getsa momentumkickinthe directionofthe laser,andwhenit spontaneouslyemits the photon,itgets amomentum kick in a randomdirection. Spontaneous emissionis the inherent source of noise in our scheme, so we explain how to represent it with the noise term ψA in Eq. (23). There are two factors that must be taken into account. First, for the experimental conditions assumed in the text, an ion scatters on the order of one photon per oscillation cycle. Thus, the noise is a sequence of occasional impulses happening at random times. Second, the noise is position dependent due to the intensity gradient of the red beams. Wejustconsiderasingleion,sincethenoiseforeachionisindependentandidenticallydistributed. Eachscattering eventhappens ata randomtime, andthe spontaneous emissionofa photoncauses a momentumkick ~k ina random direction. Suppose the ion scatters photons at times t . Then the noise in Eq. (3) of the main text is n ~k χ(t) = δ(t t )q , (24) n n m − Xn where q is a random variable (with variance σ ) for the projection of a momentum kick along the trap axis. Each n q kickisindependent( q q =δ ). Forsimplicity,weassumethattheemissionisisotropic(σ2 =1/3),althoughthere h j ki jk q is a slight anisotropy relative to the laser direction [24]. With the assumptions on experimental parameters given in the text, the scattering rate Γ may be calculated rigorously from the Optical Bloch Equations [5, 25], γ3 I (x) I R B Γ(x) = + . (25) I (cid:20)γ2+4∆ω2 γ2+4∆ω2 (cid:21) s R B The first and second terms correspond to scattering by the red and blue beams, respectively. Note that Γ depends on position and is independent of velocity to first order. After rescaling (τ =ω t and y =x/ℓ) to get Eq. (15), the noise is o ~k ζ(τ) = δ(τ τ )q , (26) n n mω ℓ − o Xn and the scattering rate becomes, Γ˜(y) = Γ˜ (y)+Γ˜ (27) R B 1 y 2 I γ3 Γ˜ (y) = B (28) R ω (cid:18)cosφ(cid:19) I γ2+4∆ω2 o s 1 I γ3 Γ˜ = B , (29) B ω I γ2+4∆ω2 o s where we have used the intensity relation given in Eq. (2) of the main text. To calculate the amplitude noise ξr, plug Eq. (26) into Eq. (19), ~k ξr(τ) = q sin(τ +θ), (30) n n ∆τmω ℓ o τ<τnX<τ+∆τ where τ is the time of a scattering event. Since the damping is weak, the ion scatters on the order of one photon in n an oscillation cycle (Γ ω /2π), so there is significant time between scattering events. This means that the phase of o ∼ oscillation at which a scattering event occurs is approximately uncorrelated with the phase of the next event. Each scattering event has a random projection and phase. Thus, the sum in Eq. (30) is over independent samples of the random variable w =q u , where u =sinτ (ignoring the unimportant phase offset θ for now). n n n n n Now we find w ’s distribution ρ . The intensity gradient of the red beams causes them to scatter more at certain n w phases within a cycle, while the blue beams scatter uniformly. Thus, ρ is actually a weighted average of red and w blue components. First we find the distribution of scattering times τ (mod 2π) from the intensity profiles, n 1 cos2τ red π ρ (τ) =  , (31) τ  1 blue 2π  7 since y =r cosτ. Thus the distribution of u =sinτ is ′ n n dτ ρ (u) = ρ (32) u τ(cid:12)du(cid:12) (cid:12) (cid:12) (cid:12)(cid:12)2√1(cid:12)(cid:12) u2 red π − =  , (33)  1 1 blue π√1 u2 −  for u 1. Since we assume isotropic spontaneous emission, the distribution of the projection q is ρ (q) = 1/2 for n q | |≤ q 1. Then the distribution of w =q u is n n n | |≤ 1 1 ρ (w) = du dq ρ (u)ρ (q)δ(w uq) (34) w u q Z Z − 1 1 − − 2 √1 w2+log1+√1 w2 red =  π h− − |w−| i , (35)  1 log1+√1 w2 blue π w−  | | for w 1. The variance of w is n | |≤ 1 red 12 σ2 =  . (36) w  1 blue 6  To find the phase noise ξθ, plug Eq. (26) into Eq. (20), ~k ξθ(τ) = q cos(τ +θ), (37) n n r ∆τmω ℓ ′ o τ<τnX<τ+∆τ and go through the same process to find the variance of v =q cosτ , n n n 1 red 4 σ2 =  . (38) v  1 blue 6  Althoughw andv comefromthe samescatteringevent,theyarestatisticallyuncorrelatedbecause sinτ cosτ = n n n n h i 0. We let the time interval ∆τ be large enough to include many scattering events but smaller than the characteristic time scales in Eqs. (17) and (18). We average the scattering rate Γ˜(y) in Eq. (27) over ∆τ to find the time-averaged scattering rates of the red beams (Γ¯ ) and blue beams (Γ¯ ), R B 1 r 2 I γ3 Γ¯ (r ) = ′ B (39) R ′ 2ω (cid:18)cosφ(cid:19) I γ2+4∆ω2 o s 1 I γ3 Γ¯ = B . (40) B ω I γ2+4∆ω2 o s Γ¯ depends on r due to the intensity gradient of the red beam. Then the amplitude and phase noises are Gaussian R ′ and described by, ~k 2 1 1 ξr(τ)ξr(τ ) = Γ¯ + Γ¯ δ(τ τ ) (41) ′ R B ′ h i (cid:18)mω ℓ(cid:19) (cid:18)12 6 (cid:19) − o 1 ~k 2 1 1 hξθ(τ)ξθ(τ′)i = r2 (cid:18)mω ℓ(cid:19) (cid:18)4Γ¯R+ 6Γ¯B(cid:19)δ(τ −τ′). (42) ′ o They are uncorrelated with each other: ξr(τ)ξθ(τ ) =0. ′ h i 8 After rescaling (t¯=ντ/2, r =r /2) to get Eqs. (21) and (22), the noises become, ′ 1 ~k 2 1 1 ψr(t¯)ψr(t¯) = Γ¯ + Γ¯ δ(t¯ t¯) (43) ′ R B ′ h i 2ν (cid:18)mω ℓ(cid:19) (cid:18)12 6 (cid:19) − o 1 ~k 2 1 1 hψθ(t¯)ψθ(t¯′)i = 2νr2 (cid:18)mω ℓ(cid:19) (cid:18)4Γ¯R+ 6Γ¯B(cid:19)δ(t¯−t¯′). (44) o Again, ψr(t¯)ψθ(t¯) =0. Finally, the complex-valued noise in Eq. (23) is, ′ h i ψA(t¯,A) = [ηR(t¯)+iσR(t¯)]A+[ηB(t¯)+iσB(t¯)] (45) H ηR(t¯)ηR(t¯) = δ(t¯ t¯) (46) ′ ′ h i cos2φ − 3H σR(t¯)σR(t¯) = δ(t¯ t¯) (47) ′ ′ h i cos2φ − ηB(t¯)ηB(t¯) = σB(t¯)σB(t¯) =Hδ(t¯ t¯) (48) ′ ′ ′ h i h i − ~(γ2+4∆ω2) H = , (49) 96mω2ℓ2∆ω o where H is a measure of the noise, and we have simplified using Eq. (4) of the main text. The noise functions for the red beams (ηR,σR) and blue beams (ηB,σB) are all uncorrelated with each other. The noise from the red beams increases with amplitude and causes more phase noise than amplitude noise.