Sympathetic EIT laser cooling of motional modes in an ion chain
Y. Lin, J. P. Gaebler, T. R. Tan, R. Bowler, J. D. Jost, D. Leibfried, D. J. Wineland
SSympathetic EIT laser cooling of motional modes in an ion chain
Y. Lin, ∗ J. P. Gaebler, T. R. Tan, R. Bowler, J. D. Jost, † D. Leibfried, and D. J. Wineland
National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA
We use electromagnetically induced transparency (EIT) laser cooling to cool motional modes of alinear ion chain. As a demonstration, we apply EIT cooling on Mg + ions to cool the axial modes ofa Be + - Mg + ion pair and a Be + - Mg + - Mg + - Be + ion chain, thereby sympathetically coolingthe Be + ions. Compared to previous implementations of conventional Raman sideband cooling, weachieve approximately an order-of-magnitude reduction in the duration required to cool the modesto near the ground state and significant reduction in required laser intensity. One proposal for building a quantum information pro-cessor is to use trapped, laser-cooled ions [1–3], whereinternal states of the ions serve as individual qubits thatare manipulated by laser beams and/or microwave radi-ation. The Coulomb coupling between ions establishesnormal modes of motion; transitions involving both thequbit states and motional modes enable entangling gateoperations between multiple qubits. For high-fidelity de-terministic entangling gates, we require that the thermalor uncontrolled components of the relevant modes be inthe Lamb-Dicke regime [2], where the amplitude of theions’ uncontrolled motion is much less than the effectivewavelength of the coupling radiation [4]. For most exper-iments this means that the motion must be cooled to nearthe quantum-mechanical ground state, which has typi-cally been achieved with sideband laser cooling [2, 5, 6].Scaling can potentially be achieved by storing ions inmulti-zone arrays where information is moved in the pro-cessor by physically transporting the ions [7, 8] or tele-porting [9].Ion motion can be excited by ambient noisy electricfields and/or during ion transport [7]. Therefore, forlengthy algorithms, a method for recooling the ions isneeded. This can be accomplished by combining thequbit ions with “refrigerant” ions that are cooled with-out disturbing the qubit states, but “sympathetically”cool the qubits through Coulomb coupling [7, 8, 10–15].Demonstrations of this technique in information process-ing have so far used sideband cooling [10–14]. While ef-fective, sideband cooling can typically cool only one modeat a time, due to the differences in mode frequencies andnarrowness of the sideband transitions. Furthermore, inthe case of stimulated-Raman transition sideband cool-ing [6], the laser-beam intensities and detuning must besufficiently large to avoid heating from spontaneous emis-sion. Importantly, in experiments performed in this scal-able configuration, the time required for re-cooling hasbeen the limiting factor [12, 16] and leads to errors dueto qubit dephasing [17]. A technique that can mitigatethese problems is EIT laser cooling, described theoreti-cally in [18, 19] and demonstrated on a single ion in [20–22]. For EIT cooling, required laser intensities are rela-tively small and the cooling bandwidth is large enoughthat multiple modes can be cooled simultaneously. To demonstrate these features, we investigate EIT coolingof multiple modes of linear ion chains containing Be + and Mg + ions. EIT cooling is applied to the Mg + ions, which cools all modes along the axis of the chainto near the ground state, thereby sympathetically cool-ing the Be + ions. We realize significant reductions incooling duration and required laser intensity comparedto previous experiments that employed sideband cooling[11–14].Following [19], consider the three-level Λ system com-prised of the bare states | g (cid:105) , | g (cid:105) and | e (cid:105) shown in Fig.1(a). For an ion at rest, laser beams with resonant Rabirates Ω and Ω and equal detunings ∆ = ∆ ≡ ∆ > | ψ D (cid:105) = (Ω | g (cid:105) − Ω | g (cid:105) ) / Ω withΩ ≡ (cid:112) Ω + Ω . Absorption from a weak (third) probelaser beam has a spectrum indicated in Fig. 1(b). Thefrequency shift between the absorption null and the rel-atively narrow peak on the right is δ = ( (cid:112) ∆ + Ω − ∆) / . (1)If the difference in k-vectors for the two dressing beamshas a component along the direction of a motional mode,the ion’s motion will prevent it from being in the darkstate. In the ion’s frame of reference, the laser beams ap-pear to be frequency modulated at the mode frequency ω . For small amplitudes of motion such that the ion is inthe Lamb-Dicke regime, the ion is probed by sidebandsat frequencies ∆ ± ω . If conditions are such that δ (cid:39) ω ,the upper sideband is resonant with the narrow featureon the right side of Fig. 1(b) and the ion can scatter aphoton while simultaneously losing one quantum of mo-tion, similar to more conventional sideband cooling. Oneadvantage of this scheme is that the width of the right-hand peak can be made broad enough that the conditionfor cooling is met for multiple modes for the same valueof δ . This may prove advantageous in experiments in-volving many ions, such as simulations where the modefrequencies have a relatively narrow distribution [23–26].We trap Be + and Mg + ions in a linear radio-frequency Paul trap described in [14], depicted schemat-ically in Fig. 1(c). The ions form a linear chain alongthe axis of the trap, the axis of weakest confinement.We perform experiments on either a single Be + - Mg + a r X i v : . [ phy s i c s . a t o m - ph ] M a r Figure 1. (a) Relevant energy levels for Mg + . The threelevels | g (cid:105) , | g (cid:105) and | e (cid:105) serve as a Λ system for EIT cooling.Laser beams with σ + and π polarizations couple the groundstates to the excited state with Rabi rates Ω and Ω anddetuning ∆. Wavy lines show spontaneous emission from theexcited state to the ground states and the excited-level decayrate is denoted with γ (cid:39) π ×
41 MHz. The fourth level | e (cid:48) (cid:105) can perturb the EIT cooling when the π polarized laser beamhas frequency near the | g (cid:105) to | e (cid:48) (cid:105) resonance. (b) Simulationof the absorption spectrum of a stationary ion by a weak probebeam for ∆ = 2 π × / π = 30 MHz, and Ω / π = 12 MHz. For simplicity, the fourth level | e (cid:48) (cid:105) is ignored for(b). The probe detuning from the | g (cid:105) to | e (cid:105) resonance isdenoted by ∆ P . This Fano-like profile contains a narrow andbroad feature corresponding to dressed states | ψ + (cid:105) and | ψ − (cid:105) respectively [19]. When ∆ P = ∆, absorption vanishes due tocoherent population trapping. (c) Beam configuration and adepiction of the Be + − Mg + − Mg + − Be + ion chain. pair or a four-ion chain with the ions in the order Be + - Mg + - Mg + - Be + [11, 14]. A single trapped Be + ionhas motional frequency ω z / π = 2 .
97 MHz along thetrap axis and { ω x / π, ω y / π } = { . , . } MHz, alongthe transverse directions. An internal-state quantizationmagnetic field B is applied along a direction 45 ◦ to thetrap axis (Fig. 1(c)), which breaks the degeneracy ofmagnetic sublevels of Be + and Mg + . In Fig. 1(a), m J indicates the projection of the Mg + ion’s angu-lar momentum along the direction of B. For B = 11.964mT, the energy splitting of the qubit states 2 s S / | F = 2 , m F = 1 (cid:105) and | F = 1 , m F = 0 (cid:105) of Be + isfirst-order insensitive to changes in B, leading to longcoherence times of superposition states [17].We apply two laser beams near the 3 s S / to 3 p P / transition in Mg + at approximately 280.353 nm (Fig.1(a)). These two beams are derived from the same laserand frequency shifted by acousto-optic modulators [27].As indicated in Fig. 1(c), one of the beams propagatesalong the direction of B with σ + polarization to couple | g (cid:105) to | e (cid:105) with resonant Rabi rate Ω and detuning ∆ from the excited state. The other beam has π polariza-tion and couples | g (cid:105) to | e (cid:105) with resonant Rabi rate Ω and detuning ∆ . We set ∆ = ∆ = ∆; (∆ / π canbe set to a precision of approximately 1.5 MHz.) Thedifference wave-vector of the two beams is parallel to thetrap axis. The values of Ω and Ω are determined frommeasurements of the Rabi rate for Raman carrier transi-tions and the AC Stark shift from the σ + polarized beamwhen it is detuned from resonance.We first apply Doppler cooling to Be + , which initial-izes the temperatures of the axial modes of motion tonear the Doppler limit ( (cid:39) (cid:126) γ Be / (2 k B ), where γ Be is the Be + excited-state decay rate and k B is Boltzmann’s con-stant). We then apply the EIT cooling beams to Mg + for a cooling duration t c . To determine the final meanmotional-state quantum number ¯ n of the normal modes,we compare the strength of red and blue Raman side-band transitions in the Be + ions on the | , (cid:105) → | , (cid:105) transition, using a pair of 313.220 nm laser beams [6, 28].The Be + - Mg + ion pair has two axial motionalmodes: a mode where the two ions oscillate in-phase ( I )with frequency ω I / π = 2 . O ) with frequency ω O / π = 4 . η = ∆ k z z , where z isthe ground state mode amplitude for the Mg + ion; here, η I = 0 .
294 and η O = 0 . δ (cid:39) ω cannot be satisfied for both modes simultaneously,since the mode frequencies are substantially different.We first perform EIT cooling on Mg + for 800 µ s,long enough for the system to reach equilibrium. We set∆ / π = 96 . / π = 12 . to vary δ (Eq. (1)).The minimum values of ¯ n I = 0.08(1) and ¯ n O = 0.04(1)are obtained when δ closely matches the respective modefrequency, as expected (Fig. 2(a)). We observe a (cid:39)
10% deviation of the value of δ needed for optimum cool-ing compared to the mode frequency, which can be ex-plained by additional AC Stark shifts and photon scat-tering from the π -polarized beam that couples | g (cid:105) to | e (cid:48) (cid:105) ≡ P / | m J = − / (cid:105) in Mg + (see Fig. 1(a)). Weperformed a numerical simulation of the full dynamicsincluding state | e (cid:48) (cid:105) . We also include the effects of heat-ing rates of both modes, ˙¯ n I = 0.38 quanta/ms and ˙¯ n O =0.06 quanta/ms. The average occupation numbers fromthe simulation are shown as solid lines in the Fig. 2(a),and are in good agreement with our experimental results. Figure 2. Mean motional excitation number ¯ n for the I (redtriangles, ω/ π = 2.1 MHz) and O (blue squares, ω/ π =4.5 MHz) axial modes of a Be + - Mg + ion pair. (a) ¯ n after800 µ s of EIT cooling as a function of δ/ π . Optimal coolingfor each mode occurs when δ approximately equals the modefrequency. (b) ¯ n plotted as a function of EIT cooling duration t c . From 0 to 74 µ s, δ (cid:39) ω O . From 75 to 85 µ s, δ (cid:39) ω I (see text). In both figures, error bars represent statisticaluncertainty of the sideband amplitude ratios. The solid linesare simulations of the full dynamics including the | e (cid:48) (cid:105) level in Mg + , measured ambient heating rates, detuning and beamintensities. In the simulations we truncated the motion tothe first 6 Fock states for both modes for the steady-statesimulation in (a) and to the first 10 (6) Fock states for I ( O )mode for temporal simulation in (b). For the simulations, we use the treatment of [19], valid inthe Lamb-Dicke regime, adjusted for the relevant modesand mode amplitudes of the Mg + ions.To investigate the temporal dynamics of the coolingwe set δ to be near a mode frequency and measure ¯ n vs.cooling duration t c . We first Doppler-cool both modeswith Be + reaching ¯ n I ∼ n O ∼
2. We find that atthe experimentally determined optimum values of δ/ π of 2.55(5) MHz and 4.87(5) MHz, the 1 /e cooling time Figure 3. Minimum ¯ n values for each of the four axial modesof a Be + - Mg + - Mg + - Be + ion chain as a function of δ/ π after 800 µ s of cooling to ensure steady state. Modes 1 to 4in the text are labeled as red triangles, green squares, blackcircles and blue diamonds, respectively. for the I mode is 4(1) µ s and for the O mode is 15(1) µ s.The faster cooling rate for the in-phase mode is expectedbecause of its larger Mg + Lamb-Dicke parameter. Wecan take advantage of the difference in equilibration timesto efficiently cool both modes, as shown in Fig. 2(b). Wefirst set δ (cid:39) ω O and apply cooling for 75 µ s, yielding¯ n O = 0 . I mode is cooled to¯ n I = 0 . δ (cid:39) ω I and apply the coolingbeams for an additional 10 µ s, reaching ¯ n I = 0 . O mode begins to heat toits equilibrium value of ¯ n = 0 . δ . However in 10 µ s, this heating is small,leading to a final value of ¯ n O = 0 . n values in 85 µ s.We also investigate sympathetic EIT cooling for thefour-ion chain Be + - Mg + - Mg + - Be + . We label thefour-ion axial modes { , , , } , which have mode fre-quencies (cid:39) { . , . , . , . } MHz and corresponding Mg + Lamb-Dicke parameters { . , . , . , . } .Fig. 3 shows the final ¯ n of each mode vs. δ/ π after800 µ s of cooling to ensure steady state. We set ∆ / π =96.7 MHz, Ω / π = 9.6 MHz, and scan Ω / π from 17to 73 MHz. The EIT cooling bandwidth is sufficient thatmodes 2, 3, and 4 can be simultaneously cooled to neartheir minimum ¯ n < .
15 by setting δ/ π = 6.1 MHz; how-ever, at this value, mode 1 is cooled only to ¯ n = 0 . µ s ofcooling with δ/ π = 6.1 MHz to cool modes 2, 3, and 4followed by 5 µ s of cooling with δ/ π = 2.4 MHz to coolmode 1, reaching ¯ n = { . , . , . , . } .In our experiments, laser beam power of the pi(sigma)-polarized beam ranged between 3 and 10 µ W (3 and17 µ W). In previous implementations of sequential Ra-
Figure 4. (a) Minimum values of ¯ n for the two-ion O mode vs.∆ / π . The peak near 223 MHz results from resonant scatter-ing on the | g (cid:105) ↔ | e (cid:48) (cid:105) transition from the π polarized light.Blue circles are the experimental data and red triangles aresimulations based on [19]. Green diamonds are simulationsnot including | e (cid:48) (cid:105) and the black solid line shows ¯ n = ( γ/ [18]. (b) Simulation of 1/e cooling time vs. the mass of ionX + that is sympathetically cooled by Mg + (axial modes),with fixed trap potential such that ω I(O) / π = 2.1 (4.5) MHzwhen X + is Be + . Red triangles are for the I mode; bluesquares for the O mode. Optimum values of δ were chosenfor each mode and ion-mass combination, with Ω / π = 5.9MHz and ∆ / π = 96.7 MHz. (Here | e (cid:48) (cid:105) is neglected) man sideband cooling [11–13, 16], cooling of these modesfrom Doppler temperatures to ¯ n ∼ . ∼ µ s with approximately an order of mag-nitude higher laser intensities.The | g (cid:105) to | e (cid:48) (cid:105) transition frequency is 223.3 MHzhigher than that of the | g (cid:105) to | e (cid:105) transition. Thus, EITcooling will be strongly affected for ∆ near 223.3 MHzdue to recoil from scattering on the | g (cid:105) to | e (cid:48) (cid:105) transi-tion. To illustrate the effect, we measure the minimumvalue of ¯ n for cooling the O mode of the Be + − Mg + ion pair as a function of the detuning ∆ (Fig. 4(a)). Foreach value of ∆ we optimize the EIT cooling by varying δ . The height of this recoil peak depends on the Rabirate ratio Ω / Ω with higher ratios leading to a highervalues of ¯ n . For data of Fig. 4(a) the ratio was held at0.24. We note that for large detuning ∆, higher laserintensity is needed to maintain values of δ near the modefrequencies.When ∆ k z is aligned along the trap axis, the mo-tional modes along the transverse axes are heated byphoton recoil. To study this effect we first cool one ofthe transverse modes of a Be + - Mg + pair (frequency (cid:39) Be + . We then apply an EIT coolingpulse on the O mode, with similar laser beam conditionsas above. After 60 µ s the O mode is cooled from theDoppler temperature (¯ n (cid:39)
2) to ¯ n = 0.04(1) while thetransverse mode is heated from ¯ n = 0 . . O mode is cooled to near its minimum value,the heating rate of the transverse mode decreases becausethe ion becomes approximately trapped in the dark statefor spin and the ground state of axial motion. This rela-tively low transverse excitation should cause a negligibleerror on a two-qubit gate, which is affected by the trans-verse modes only through second-order coupling to theaxial mode frequencies [29]. Furthermore, Doppler cool-ing of all modes before EIT cooling would prohibit anycumulative effect of the heating in experiments requiringmany rounds of sympathetic cooling.To study the efficiency of EIT sympathetic cooling onother ion species, such as Al + , Ca + and Yb + etc.,we simulate cooling of an ion pair Mg + - X + , whereX + is the sympathetically cooled ion of different mass,as shown in Fig. 4(b). Smaller differences in ion masslead to more balanced mode amplitudes and a reductionin the difference of cooling rates for individual modes.Large mass imbalances lead to at least one motional modehaving a small Mg + amplitude and thus a long coolingtime [15].In summary, we have described sympathetic coolingof Be + ions by EIT cooling of Mg + ions held in thesame trap. We investigate the cooling for both an ionpair and a four-ion chain crystal that can be used asa configuration for performing entangling gates betweenpairs of Be + ions in a scalable architecture [11–13]. Bytaking advantage of the different cooling rates for differ-ent modes of motion we demonstrated a two-stage EITcooling scheme that can bring all modes to near theirminimum excitation level. Compared to previous im-plementations of conventional Raman sideband cooling,sympathetic EIT cooling provides a broad cooling band-width, requires less laser power, and is technically easierto implement. This method may also be useful for sym-pathetic cooling of molecular ions, for use in quantumlogic spectroscopy [10], trapped-ion quantum simulation[23–26], strongly-confined neutral atoms [30], and nano-mechanical resonators [31].We thank C. W. Chou and R. J¨ordens for helpful dis-cussion on simulation. Also we thank J. J. Bollinger, Y.Colombe, G. Morigi and T. Rosenband for helpful com-ments on the manuscript. This work was supported byIARPA, ARO contract No. EAO139840, ONR and theNIST Quantum Information Program. J. P. G. acknowl-edges support by NIST through an NRC fellowship. Thispaper is a contribution by NIST and is not subject to U.S.copyright. ∗ Electronic address: [email protected] † Current address: ´Ecole Polytechnique F´ed´erale de Lau-sanne, Lausanne, Switzerland[1] J. I. Cirac and P. Zoller, Phys. Rev. Lett. , 4091 (1995).[2] R. Blatt and D. Wineland, Nature , 1008 (2008). [3] R. Blatt and C. F. Roos, Nat. Phys. , 277 (2012).[4] Probabilistic schemes for entanglement do not requiresuch strong confinement; see C. Monroe, R. Raussendorf,A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, andJ. Kim, arXiv:1208.0391.[5] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J.Wineland, Phys. Rev. Lett. , 403 (1989).[6] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano,and D. J. Wineland, Phys. Rev. Lett. , 4714 (1995).[7] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried,B. E. King, and D. M. Meekhof, J. Res. Natl. Inst. Stand.Technol. , 259 (1998).[8] D. Kielpinski, C. Monroe, and D. J. Wineland, Nature , 709 (2002).[9] D. Gottesman and I. L. Chuang, Nature , 390 (1999).[10] P. O. Schmidt, T. Rosenband, C. Langer, W. M. Itano,J. C. Bergquist, and D. J. Wineland, Science , 749(2005).[11] J. D. Jost, J. P. Home, J. M. Amini, D. Hanneke, R.Ozeri, C. Langer, J. J. Bollinger, D. Leibfried, and D. J.Wineland, Nature , 683 (2009).[12] J. P. Home, D. Hanneke, J. D. Jost, J. M. Amini, D.Leibfried, and D. J. Wineland, Science , 1227 (2009).[13] D. Hanneke, J. P. Home, J. D. Jost, J. M. Amini, D.Leibfried, and D. J. Wineland, Nat. Phys. , 13 (2009).[14] J. D. Jost, Ph.D. thesis, University of Colorado, Boulder,2010.[15] J. B. W¨ubbena, S. Amairi, O. Mandel, and P. O.Schmidt, Phys. Rev. A , 043412 (2012).[16] J. P. Gaebler, A. M. Meier, T. R. Tan, R. Bowler, Y. Lin,D. Hanneke, J. D. Jost, J. P. Home, E. Knill, D. Leibfried,and D. J. Wineland, arXiv:1203.3733 and Phys. Rev.Lett. , 260503 (2012).[17] C. Langer, R. Ozeri, J. D. Jost, J. Chiaverini, B. De-Marco, A. Ben-Kish, R. B. Blakestad, J. Britton, D. B.Hume, W. M. Itano, D. Leibfried, R. Reichle, T. Rosen-band, T. Schaetz, P. O. Schmidt, and D. J. Wineland, Phys. Rev. Lett. , 060502 (2005).[18] G. Morigi, J. Eschner, and C. H. Keitel, Phys. Rev. Lett. , 4458 (2000).[19] G. Morigi, Phys. Rev. A , 033402 (2003).[20] C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J.Eschner, and R. Blatt, Phys. Rev. Lett. , 5547 (2000).[21] F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D.Leibfried, A. Mundt, and R. Blatt, Appl. Phys. B ,807 (2001).[22] S. Webster, Ph.D. thesis, University of Oxford, 2005.[23] G.-D. Lin, S.-L. Zhu, R. Islam, K. Kim, M.-S. Chang, S.Korenblit, C. Monroe, and L.-M. Duan, Europhys. Lett. , 60004 (2009).[24] R. Islam, E. E. Edwards, K. Kim, S. Korenblit, C. Noh,H. Carmichael, G.-D. Lin, L.-M. Duan, C.-C. J. Wang,J. K. Freericks, and C. Monroe, Nat. Commun. , 377(2011).[25] B. C. Sawyer, J. W. Britton, A. C. Keith, C.-C. J. Wang,J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger,Phys. Rev. Lett. , 213003 (2012).[26] R. Islam, C. Senko, W. C. Campbell, S. Korenblit, J.Smith, A. Lee, E. E. Edwards, C.-C. J. Wang, J. K. Fre-ericks, and C. Monroe, arXiv:1210.0142.[27] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts,W. M. Itano, D. J. Wineland, and P. Gould, Phys. Rev.Lett. , 4011 (1995).[28] B. E. King, C. S. Wood, C. J. Myatt, Q. A. Turchette, D.Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland,Phys. Rev. Lett. , 1525 (1998).[29] C. F. Roos, T. Monz, K. Kim, M. Riebe, H. H¨affner,D. F. V. James, and R. Blatt, Phys. Rev. A , 040302(2008).[30] T. Kampschulte, W. Alt, S. Manz, M. Martinez - Do-rantes, R. Reimann, S. Yoon, D. Meschede, M. Bienert,and G. Morigi, arXiv:1212.3814 (2012).[31] K. Xia and J. Evers, Phys. Rev. Lett.103