Ion transport and reordering in a two-dimensional trap array
Y. Wan, R. Jördens, S. D. Erickson, J. J. Wu, R. Bowler, T. R. Tan, P.-Y. Hou, D. J. Wineland, A. C. Wilson, D. Leibfried
IIon transport and reordering in a two-dimensional trap array
Y. Wan,
1, 2, ∗ R. J¨ordens,
1, 2, † S. D. Erickson,
1, 2
J. J. Wu,
1, 2
R. Bowler,
1, 2, ‡ T.R. Tan,
1, 2, § P.-Y. Hou,
1, 2
D. J. Wineland,
1, 2, 3
A. C. Wilson, and D. Leibfried National Institute of Standards and Technology, Boulder, CO 80305, USA Department of Physics, University of Colorado, Boulder, CO 80309, USA Department of Physics, University of Oregon, Eugene, OR 97403, USA (Dated: March 10, 2020)Scaling quantum information processors is a challenging task, requiring manipulation of a largenumber of qubits with high fidelity and a high degree of connectivity. For trapped ions, this could berealized in a two-dimensional array of interconnected traps in which ions are separated, transportedand recombined to carry out quantum operations on small subsets of ions. Here, we use a junctionconnecting orthogonal linear segments in a two-dimensional (2D) trap array to reorder a two-ioncrystal. The secular motion of the ions experiences low energy gain and the internal qubit levelsmaintain coherence during the reordering process, therefore demonstrating a promising method forproviding all-to-all connectivity in a large-scale, two- or three-dimensional trapped-ion quantuminformation processor.
INTRODUCTION
Coherent manipulation of trapped atomic ions enablesapplications ranging from quantum sensing (e.g. forceand field sensing, precision spectroscopy, optical clocks)to quantum information processing. Most applicationsmust deal with the difficulty in controlling multiple ions,where for optical clocks the number of clock ions lim-its the frequency stability [1], and for quantum infor-mation processing the number of qubits limits the pro-cessing capability. One approach to increase the num-ber of ions in a system is to confine individual ions orsmall groups of ions in separate trap zones of an array.Ions are then connected either through probabilistic ion-photon coupling [2, 3], or as discussed here, by employ-ing the “quantum charge-coupled device” (QCCD) archi-tecture [4, 5], in which ions are transported throughoutthe array to provide the high connectivity required forefficient implementation of general algorithms. This ap-proach can extend the features of small ion crystals, suchas high-fidelity quantum gates [6, 7] and precise prepa-ration and characterization of the motional state [8], toa larger number of qubits.One of the key elements of the QCCD architecture isthe ability to reconfigure ion crystals and to hold subsetsof ions in different locations, ensuring mutual isolation,while operating on them in parallel. High connectivityand parallelism are considered to be crucial for large-scale fault-tolerant quantum computation [9]. This re-quires separation, transport, and rearrangement of ionsthroughout multiple trapping zones. Previous experi-ments have demonstrated adiabatic transport of bothsingle ions [10] and chains of ions [11], diabatic trans-port and separation [12, 13], and fast swapping of neigh-boring ions in a one-dimensional (1D) array by rotatingtwo ions in place [14–16]. These primitives have enableda transport-based quantum logic gate [17, 18], scalablecreation of multi-partite entanglement with bipartite in- teractions [19], quantum-state-assisted sensing [20], testsof local realism [21], and quantum gate teleportation [22].Specific to developing a trapped-ion quantum com-puter, a multi-dimensional trap array is desired to fullyrealize the power of the QCCD architecture, in whichmultiple linear trap segments are connected by junctions[10, 11, 23–27]. Such a multi-dimensional trap arrayenables smaller average distances between ions than inlower-dimensional architectures and efficiently extendsthe all-to-all coupling between ions in a small chain [28] toconnecting arbitrary subsets of a larger number of qubits.Previous work in traps featuring a junction demon-strated the low-temperature shuttling of ions through RFjunctions and characterized the resulting kinetic energyincrease [10, 11]. Ion crystal reconfigurations were re-ported in [23, 27] (using junctions but without measure-ment of kinetic energy changes) or using crystal rotationin a 1D architecture [14–16]. Here, we use a junction todistribute Be + ions in a two-dimensional (2D) architec-ture and reorder two ions initially in the same potentialby combining adiabatic transport and separation primi-tives. We show that the coherence in the internal statesof the ions is maintained during the process and charac-terize excess motional excitation. EXPERIMENTAL SETUP
A schematic of the trap array is shown in Fig. 1a. Thephysical structure consists of two electrode wafers sepa-rated by 250 µ m, providing strong confinement along allthree directions, and a third layer (500 µ m below the bot-tom layer) possessing one single electrode that serves asa common bias electrode across the entire trap. The trapfeatures three linear regions along the − z , + z , and + x di-rections (containing zones S , H , and V respectively) withthe origin located at the center of the RF junction ( C )in a plane that is parallel to, and in the middle between a r X i v : . [ qu a n t - ph ] M a r
500 0 500 z ( m)5002500250500 x ( m ) CS HV rf bumps rf bumps 10 P s e u d o - p o t e n t i a l ( e V ) b ion a S V HC top view
B RAL x-y cross section y FIG. 1.
Schematic of the X-junction trap. (a)
Schematic view of the top wafer of the trap, with DC (con-trol) electrodes in orange and RF electrodes in blue. A secondwafer below the top wafer has DC and RF electrodes swapped,as indicated in the cross-section. The ion shown is located onthe axis x = y = 0 of a linear portion of the trap. The ionsare held in three major experiment zones, labeled by S , H ,and V , connected by the junction located at C . Trappingzones L , A , B , and R lying in the same linear region as S are used together with the zone S to perform operations suchas separation, recombination, and individual addressing anddetection. (See text for more details.) (b) Pseudo-potentialalong the linear channels connected by the junction in theplane equidistant to the two wafers defined to be y = 0. Thejunction gives rise to four pseudo-potential bumps (indicatedby blue arrows) around C . the two electrode wafers. All other relevant zones ( L , A , B , R ) for this work are marked in Fig. 2. The X-shapedjunction (X-junction) at C allows us to route ions to S , H , and V . Deviation from an ideal infinitely-long linearPaul trap gives rise to pseudo-potential “bumps” around C along the x - and z -directions as illustrated in Fig. 1b.Besides complicating ion transport, this has further im-plications for quantum information experiments, such asa position-dependent RF modulation index [29] which af-fects laser operations and position-dependent qubit fre-quencies from trap-RF-induced AC-Zeeman shifts. These pseudo-potential bumps can also introduce additionalmotional heating originating from noise in the RF gra-dients [10]. More details on the trap can be found in[10, 11, 30].All laser beams for coherent manipulation, state prepa-ration, and state detection are focused to S with waistsof approximately 25 µ m, while ions in nearby trappingzones are at least 390 µ m away during illumination ofions in S . We encode qubits in first-order magnetic-field-insensitive hyperfine ground states of Be + ions, with | F = 1 , m F = 1 (cid:105) ≡ |↑(cid:105) and | , (cid:105) ≡ |↓(cid:105) [31]. Prior tostate detection, |↑(cid:105) is transferred to | , (cid:105) ≡ | Bright (cid:105) and |↓(cid:105) to | , − (cid:105) ≡ | Dark (cid:105) . A resonant laser driving the S / | , (cid:105) ↔ P / | , (cid:105) cycling transition is used to dis-tinguish the two states through photon counts [32–34].The motional heating and excess energy accumulatedduring certain transport primitives is investigated by run-ning the test sequences listed in Table I. To characterize aprimitive, a single Be + ion or two Be + ions are initial-ized in S by cooling the axial modes ( ω COM = 2 π × . ω STR = 2 π × . n = 0 . n COM = 0 . n STR = 0 . n of approximately0 .
5. After completing a transport test sequence and re-turning to S , the final state of the ion motion is probedwith motion-sensitive Raman transition beams on blueand red sidebands [35, 36] in separate experiments. As-suming a thermal distribution of final energies, we extract¯ n of the ion motional modes by fitting a Rabi-oscillationmodel to the data from both experiments. More detailsof the motional state analysis are provided in the supple-mentary materials.We now describe several of the experimentally imple-mented transport primitives in more detail (see also Ta-ble I for a summary). We trap two Be + ions ( a and b )in a single well S and subsequently separate them froman initial spacing of about 5 µ m into two individual wellslocated at A and B . The well minima are separated by ∼ µ m and are formed in ∼ µ s by ramping the har-monic and quartic terms of the potential [37]. A suitablestatic electric field along the axial direction of the crystal,superimposed on the separation waveform, shifts the cen-ter of the ion crystal relative to the center of the quarticpotential, enabling control over the number of ions trans-ported into the respective individual wells [12]. Whenseparating two ions into wells A and B , this primitiveis denoted as S ab → A a B b in Table I (row 6). Here, thesymbols before and after the right arrow denote the initialand final configuration of the primitive, respectively. Thecapital letters of a configuration denote the positions ofthe wells, and the subscripts denote the ion(s) residing ineach well, respectively. Each ion travels a distance of ap-proximately 167 . µ m. To characterize S ab → A a B b , we a b ssssss t i m e SA BL R
FIG. 2.
Individual addressing and detection sequence.
Using the sequence depicted within the dashed lines, ions ineach well can be manipulated and detected individually, withthe shuttling operations in the dashed box taking approxi-mately 1 ms. Ion a and b are first separated into zones A and B , and then a is shuttled into S while b moves to zone R . This formation allows for internal state manipulation anddetection of a . When a is shifted to L , ion b enters S and canbe manipulated and/or its state detected. run a longer test sequence S ab → A a B b → S ab , which isimplemented by concatenating the forward and reversedversion of the primitive S ab → A a B b , and measure the ex-citation in the center-of-mass (COM) and stretch (STR)mode after recombination of a and b in S (the well inwhich the motional state is characterized at the end of atest sequence is underlined in Table I). We find an aver-age COM mode occupation of ¯ n =0.55(3) and 0.43(3) inthe STR mode.An important shuttling sequence, illustrated in thedashed box in Fig. 2, will be referred to as the “indi-vidual addressing and detection sequence”: A i B j → S i R j → A i B j → L i S j → A i B j . (1)Additional laser pulses can be applied during configu-rations S i R j and L i S j to manipulate the ions’ internalstates individually after separation and before detection(Fig. 2). This allows us to individually rotate each ion onits qubit Bloch sphere. The total duration of the trans-port in the individual addressing and detection sequence,not including any manipulation or detection operations,is about 1 ms. This sequence can also be used to deter-mine the number of ions in each well.Combining S ab → A a B b with the individual address- ing and detection sequence allows us to probe the tem-perature of individual ions. To this end, motion-sensitiveRaman beams are applied in the configuration S a R b todetermine the temperature of the ion a in the left well,and L a S b for the ion b in the right well, deriving averageoccupation numbers of 0.10(1) and 0.25(2), respectively,using sideband thermometry [35]. REORDERING TWO IONS
By combining further transport primitives (see TableI for their individual characteristics), we demonstrate re-ordering of a two-ion crystal by separating ions a and b and then moving the ions around each other with theaid of the X-junction. This is done by separating atwo-ion crystal S ab → A a B b , shuttling of ion b to V ( A a B b → A a C b → A a V b , step I in Fig. 3a), shuttlingion a to H ( A a V b → C a V b → H a V b , step II), moving ion b to A ( H a V b → H a C b → H a A b , step III), moving ion a to B ( H a A b → C a A b → B a A b , step IV), and combin-ing a and b ( B a A b → S ba ). The full reordering sequencereads S ab → A a B b → A a C b → A a V b → C a V b → H a V b → H a C b → H a A b → C a A b → B a A b → S ba (2)with the duration of each individual segment listed inTable I. The duration of the reordering sequence withoutseparation and recombination ( A a B b to B a A b ) is about1 . A a B b or B a A b in sequence (2) to encode thespin state of each ion before reordering and to detect theion positions after reordering. The encoding is performedusing a pair of co-linear Raman beams in the configura-tions S a R b and L a S b , addressing only the ions locatedat S . To verify a position swap between the two ions,we first apply a laser pulse for various durations to Rabi-flop ion a in the configuration S a R b . After the reorderingsequence, a population oscillation as a function of pulseduration is detected only in S a L b (on ion a ), and not in R a S b (on ion b ) as shown in Fig. 3b, indicating success-ful reordering. Rabi-flopping ion b in the same fashionyields a similar result, now with b exhibiting populationoscillations and a remaining in the same state.We also show that qubit coherence is maintained afterreordering by applying a Ramsey sequence to each ionindividually. We apply a π/ a ( b ) in theconfiguration S a R b ( L a S b ), execute the reordering, andapply the second π/ π/ a ( b ) TABLE I.
Summary of transport primitives for Be + ions. To characterize the performance of each primitive, a testsequence is used where the motional excitation is measured in one well configuration (underlined) towards the end or in themiddle of the test sequence. At the beginning of the test sequence, the axial modes of ion crystals are cooled close to themotional ground states (¯ n = 0 . n COM = 0 . n STR = 0 . C and C (cid:48) here indicate the same trapping zone, but with the weakest axis of the trapping potential aligned with the z -axis and x -axis,respectively. Subtracting the initial ¯ n after ion preparation and excitations during common sections in the test sequences allowsus to derive the excess motional excitation per transport primitive ∆ n p . If a primitive is run forward and backward, we assumethat these two parts contribute equally to ∆ n p . The test sequence for determining the motional excitation of the primitive S ab → A a B b (row 6) includes mode mixing in the process of separation and recombination, therefore no values for ∆ n p arederived. As a consequence, there is no prediction of motional excitations in the well configuration A a B b , and the measurementresults in row 7-8 are used as the baseline for the test sequences in the rows below. n Duration ( µ s) Distance ( µ m) ∆ n p S a → A a S a → A a → S a Be + A a → C a S a → A a → C a → A a → S a Be + H a → C a S a → A a → C a Be + → H a → C a → A a → S a C a → C (cid:48) a S a → A a → C a Be + → C (cid:48) a → C a → A a → S a V a → C a S a → A a → C a → C (cid:48) a → V a Be + → C (cid:48) a → C a → A a → S a S ab → A a B b S ab → A a B b → S ab Be + - Be + A a B b → S a R b S ab → A a B b → S a R b Be + - Be + → A a B b → L a S b → A a B b → S ab A a B b → L a S b S ab → A a B b → S a R b Be + - Be + → A a B b → L a S b → A a B b → S ab A a B b → A a C b S ab → A a B b → A a C b → A a B b → S a R b Be + - Be + → A a B b → L a S b → A a B b → S ab A a B b → A a C b S ab → A a B b → A a C b → A a B b → S a R b Be + - Be + → A a B b → L a S b → A a B b → S ab in the reverse configuration S a L b ( R a S b ). The resultingRamsey fringes show a contrast close to 1 for both ionsaddressed as shown in Fig. 3c. The phase shift is mainlycaused by the different durations that the two ions inte-grate over a − . ≈ − . a and ≈ − . b ). The frequency difference is due to the AC-Starkshift on the qubit transition from the laser beams imple-menting the Ramsey pulses, whose frequency differenceis set to be on resonance while rotating the state of theions in the presence of the AC-Stark shift. In addition,AC-Zeeman shifts and a magnetic field gradient acrossthe trap are non-negligible when shuttling ions acrossmillimeter length scales in our setup. A separate inves-tigation shows that the AC-Zeeman shifts on the qubittransition ( |↑(cid:105) ↔ |↓(cid:105) ) of Be + vary by 10 Hz over a dis-tance of about 15 µ m along the x -direction at S . From ameasured static magnetic field gradient of approximately4 . × − T / m and the second-order field sensitivity co-efficient c = 3 . × − Hz /µ T [31], we estimate aqubit frequency shift of ≈ µ m [31, 38]. These effects will need to be minimized,properly calibrated, or reduced by dynamical decouplingor error correction in future large-scale devices.The axial temperature of each ion after reordering, measured through sideband thermometry, indicates thatthe full sequence introduces an average motional exci-tation of 1.1(1) for ion b and 1.7(1) for ion a (Fig. 3d-e). The measured motional excitation after reorderingis a factor of 2-3 times larger than the values obtainedby summing up the excitation of the constituent primi-tives. We believe that the additional excitation can beattributed to non-continuous concatenation and heatingduring idle periods in static wells, but this requires fur-ther experimental and theoretical study. POTENTIAL LARGE-SCALECONFIGURATIONS
Here, we discuss several architectures for buildingmulti-qubit quantum devices using trapped ions. Relatedarchitectures are considered in [39, 40]. a) 1D chain of individually-addressed ions(Fig. 4a).
All ions are trapped in a single well andare connected by the long-range Coulomb interaction,while individual control of each ion is realized by tightly-focused laser beams. A theoretical study determined themethod to be applicable for an arbitrarily long chain ofions [41], and experimentally, two-qubit gates betweenany two qubits within a 11-qubit ion chain have been ed b ( s)0.00.51.0 b r i g h t p o p u l a t i o n a ( s)0.00.51.0 b r i g h t p o p u l a t i o n c b r i g h t p o p u l a t i o n b a ( s)0.00.51.0 b r i g h t p o p u l a t i o n a I II IV a b ab
III b a b a b aba b aba b ~ 300 s ~ 260 s ~ 320 s ~ 240 s FIG. 3.
Reordering two Be + ions using the X-junction. (a) Schematic representation of reordering sequence. Twoions a and b in the double well potential are shuttled sequentially through the junction to separated regions of the trap array,and then moved back to the initial well with their order swapped. The arrows (orange for ion a and green for ion b ) indicatethe trajectories of each ion (light blue circles) and the blue circles represent the end points of the primitives on the trajectories. (b) Ion a is excited using a single-qubit rotation in the configuration S a R b , and a Rabi oscillation is observed in S a L b (orange)after the reordering sequence. No population oscillation is observed for detection performed in R a S b (green), while ion b isideally in S . (c) Coherence of the internal states of ion a (orange) and b (green) is maintained in two corresponding Ramseysequences enclosing an exchange of ion positions and addressing one of the ions respectively. The phase shifts (0 . a (orange) as the offset of the fringe minimum from 0 and 2 . b (green)) arise mainly from the durationsthat the two ions accumulate phase due to a frequency shift relative to the local oscillator. See text for more details. (d-e) The temperatures of the two Be + ions are probed on the red (red dots) and blue (blue dots) sidebands after the reorderingsequence. Fits to the Rabi oscillation model outlined in the supplementary material (solid lines) result in average occupationnumbers of 1.1(1) for ion b and 1.7(1) for ion a . performed [42]. b) 1D trap array (Fig. 4b). As an extension of the1D chain in a single well, one can confine the ions withina 1D array of potential wells and apply global rotationson ions in each single well. The ions in different wells arethen connected via linear shuttling and separation com-bined with swap gates or crystal rotations [15]. In sucha 1D trap array, it takes O ( n ) such rotations to trans-port an ion across n other ions using a sequence of crys-tal rotations. Considering the equivalence of re-orderingand swap operation, this will induce an overhead (longercircuit depth) on the order of O ( n ) when rearranginga quantum circuit of n qubits with all-to-all connectiv-ity to a corresponding circuit with only linear nearestneighbor interactions [43]. By confining multiple ionsin each segment, one can combine the all-to-all couplingas in architecture a) and the coupling between segments through shuttling. This method is predicted to achievefault tolerance with modest reduction of the fault toler-ance threshold compared to the 2D surface code despitethe topological restriction posed by the 1D architecture[44]. c) Multi-dimensional trap lattices (Fig. 4c). In-dividual trapping zones are held in fixed positions inspace, forming a lattice of singly-occupied confining po-tentials with nearest-neighbor couplings tuned by bring-ing the traps in and out of resonance. The higher dimen-sion provides each qubit with a larger number of nearestneighbors [45, 46]. An open electrode geometry, as for ex-ample provided by surface electrode traps [47] is advan-tageous to not substantially reduce nearest neighbor cou-pling by shielding from nearby electrodes. This increasedconnectivity between the ions therefore reduces the over-head in circuit depth for implementing a quantum circuit a bc d
FIG. 4.
Chip-based multi-qubit quantum devices. (a)
1D chain of individually-addressed ions. Tightly focusedbeams allow individual addressing of ions in the chain. (b)
1D trap array. Ions are confined in separated wells created byelectrode arrays. Separation between the wells that is large compared to the electrode dimensions allows for isolated controlof each well. Coupling of a certain pair of ions are realized by repeated swap operations or crystal rotation (black arrows) andseparation/recombination. (c)
Multi-dimensional trap lattices. Ions are confined in fixed potential wells, while the couplingbetween the ions is enabled by tuning the potentials (black arrow). (d)
Multi-dimensional trap array. Couplings between theions are realized by shuttling information carriers (blue) or messengers (orange) through dedicated sections of the array. [43]. For example, employing a two-dimensional squarelattice where each ion has four nearest neighbors alreadyreduces this overhead to O ( √ n ), while 3D square lat-tice shares a similar feature with a reduced overhead of O ( √ n ) [43]. d) Multi-dimensional trap array (Fig. 4d). Amulti-dimensional trap array could combine all modes ofoperation discussed in a)-c) with all-to-all connectivitybetween the ions by shuttling through dedicated informa-tion highways, while entangled pairs of resource ancillascan be distributed ahead of time for quantum gate tele-portation to reduce the latency required by the shuttlingprocess itself [22]. In comparison to the 1D architecture,the geometry of the trap in such an multi-dimensionalarchitecture also reduces the average distance betweenany pair of ions. Small-scale reconfiguration within lin-ear regions of the multi-dimensional array would mostefficiently be done through swap gates or crystal rotationas discussed in b), while connecting distant qubits or re-placing lost ions would involve moving through junctionsand dedicated transport highways. Such an architecturemore capable in reconfiguration of ion crystals will likelybe required for the construction of multiple logical qubits[9].
SUMMARY
By combining separation and shuttling primitives, wewere able to reorder two Be + ions using an RF junc-tion connecting three trap zones on different sides ofthat junction. We verified the reordering using transport-assisted individual addressing, and showed that quantumcoherence encoded on individual qubits was maintainedduring the reconfiguration. We also briefly discussed ex-ample configurations for large-scale devices.We thank P. Kent and J. F. Niedermeyer of NIST forhelpful comments on the manuscript. This work was sup-ported by the Office of the Director of National Intelli-gence (ODNI) Intelligence Advanced Research ProjectsActivity (IARPA), ONR, and the NIST Quantum In-formation Program. S.D.E. acknowledges support by theU.S. National Science Foundation under Grant No. DGE1650115. Y.W., S.D.E., and J.J.W. are associates in theProfessional Research Experience Program (PREP) op-erated jointly by NIST and University of Colorado Boul-der. 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SUPPLEMENTARY MATERIALSWaveform generation
We use the package bem [48] that uses the boundaryelement method to generate a potential map for each in-dividual electrode, while all other electrodes are held atground potential. This potential map includes both theeffect of electrostatic potentials from DC electrodes andthe contribution of pseudo-potentials from RF electrodes.Starting from this map, we construct the potentials ex-perienced by the ions while several electrodes are heldat non-zero potential. Vanishing first derivatives of thepotentials define the saddle points and equilibrium po-sitions for single ions, positive second derivatives definetrap frequencies, and third and fourth derivatives needto be taken into account during ion separation. We usethe package electrode [49] (which uses cvxopt [50]) to pro-duce waveforms (a set of potentials, with each memberapplied to a certain electrode as a function of time) tofulfill the full set of constraints that define a transportprimitive. The waveform simulation does not includethe Coulomb interaction between ions, so the responseof multi-ion crystals is calculated after generating thewaveform by performing a normal mode calculation [51].Before applying the waveforms to the trap electrodes,we pre-compensate the waveforms [52, 53] to account forfilter distortion from the low-pass filters used to reducetechnical noise on the trap electrodes. These waveformswith pre-compensation are generated with an arbitrarywaveform generator with 50 MHz update rate [52].
Sideband thermometry
Ion crystal temperature is determined by probing thered- and blue-sideband (RSB and BSB) transitions fol-lowed by a state-dependent fluorescence detection. For asingle ion probed along one particular mode on the 1stBSB and 1st RSB, the measurement results are fittedwith model functions for the population P remaining inthe initial state after driving the sideband for a duration t [35, 36] P BSB ( t ) = N − (cid:88) n =0 p n ·
12 [1 + cos(Ω n, +1 t ) exp( − γt )] P RSB ( t ) = p + N (cid:88) n =1 p n ·
12 [1 + cos(Ω n, − t ) exp( − γt )] , (3)where p n = ¯ n n (¯ n +1) n +1 is the population in each motionalnumber state | n (cid:105) assuming a thermal distribution with anaverage phonon number ¯ n . Ω n,κ is the Rabi frequency for | n (cid:105) on the probed sideband κ with κ = +1 for the first blue sideband and κ = − γ is aphenomenological decay constant, and N is the quantumnumber of the highest Fock state | N (cid:105) considered in themodel. The Rabi frequency Ω n,κ takes the formΩ n,κ = Ω D n,κ,η = Ω exp ( − η (cid:18) n < ! n > ! (cid:19) / η | κ | L | κ | n < ( η ) , (4)where n < and n > are the lesser and greater of n and n + κ , η is the Lamb-Dicke parameter, and L yx ( z ) is a general-ized Laguerre polynomial. The decay constant γ is in-troduced as a simplified description of the decay of Rabioscillations coming from various effects, including mo-tional dephasing, spontaneous emission, amplitude andphase noise of laser beams, and anomalous heating. Weuse this model to extract the average phonon number ¯ n after initial sideband cooling and optionally after trans-port of a single ion as shown in Fig. 3d-e.If more than one mode is involved in the probe process,meaning that Lamb-Dicke parameters for the probe laserbeam are non-zero for more than one mode, one needsto include the Debye-Waller-type coupling induced bythe spectator modes [4, 54]. This is the case, e.g., fora mixed-species two-ion crystal probed along the crystalaxis, or for a single ion probed along a direction that hasoverlap with more than one normal mode direction ofthe confining potential. We model the Rabi oscillationsdriven on a single ion, but involving two motional modesas P (1)BSB ( t ) = M (cid:88) m =0 N − (cid:88) n =0 p nm · (cid:2)
1+ cos(Ω nm, +1 0 t ) exp( − γt ) (cid:3) P (1)RSB ( t ) = M (cid:88) m =0 (cid:110) p m + N (cid:88) n =1 p nm · (cid:2) nm, − t ) exp( − γt ) (cid:3)(cid:111) P (2)BSB ( t ) = N (cid:88) n =0 M − (cid:88) m =0 p nm · (cid:2)
1+ cos(Ω nm, t ) exp( − γt ) (cid:3) P (2)RSB ( t ) = N (cid:88) n =0 (cid:110) p n + M (cid:88) m =1 p nm · (cid:2) nm, − t ) exp( − γt ) (cid:3)(cid:111) . (5)The four equations above return the population that re-mained in the initial state when probed on BSB of 1stmode, RSB of 1st mode, BSB of 2nd mode, and RSB of2nd mode, respectively, for a duration t . Here, p nm is0the population in the Fock state | n, m (cid:105) , andΩ nm,κλ = Ω D n,κ,η D m,λ,η (6)is the Rabi rate of driving a transition | n, m (cid:105) →| n + κ, m + λ (cid:105) with κ and λ indicating changes of mo-tional quantum numbers in the two modes, and η , η are the Lamb-Dicke parameters of the two modes [4].The curve fitting is performed on four data curves (firstred and blue sidebands on two modes) simultaneouslyassuming a common set of fitting parameters. In par-ticular, this means that even for a mode with ¯ n = 0Rabi oscillations on the BSB will show a decay originat-ing from the statistical distribution in motional statesof the other mode. Evaluating the Rabi oscillation datausing a common set of parameters allows for a more pre-cise determination of the average phonon number at low¯ n by considering a more complete model and also at high¯ n even if red and blue sideband excitations have onlysmall differences from the bounded decay of the compan-ion mode.To determine the temperature of two-ion crystals ofthe same species, we use a model describing the Rabioscillations of two trapped ions that are simultaneouslydriven. Here, the Rabi oscillations of two ions are con-sidered as the summation of Rabi oscillations of manythree-level systems. An analytical solution for the prob-lem describing the Rabi oscillations of a three-level sys-tem has been provided in [55]. For ions fluorescing in |↑(cid:105) and not fluorescing in |↓(cid:105) , the average fluorescence at aprobe duration t , normalized to two ions fluorescing in |↑↑(cid:105) , is P ( t ) = 1 · P |↑↑(cid:105) + 12 · P |↑↓(cid:105) + |↓↑(cid:105) , (7)and takes following the forms when probed on the BSBof the center-of-mass (COM) mode, RSB of the COMmode, the BSB of the stretch (STR) mode, and RSB of the STR mode respectively P COMBSB ( t ) = M (cid:88) m =0 N − (cid:88) n =0 p nm · (cid:0) | c (0) nm, +1 0 | + 12 · | c (1) nm, +1 0 | (cid:1) P COMRSB ( t ) = M (cid:88) m =0 (cid:110) p m + N (cid:88) n =1 p nm · (cid:0) | c (0) nm, − | + 12 · | c (1) nm, − | (cid:111) P STRBSB ( t ) = N (cid:88) n =0 M − (cid:88) m =0 p nm · (cid:0) | c (0) nm, | + 12 · | c (1) nm, | (cid:1) P STRRSB ( t ) = N (cid:88) n =0 (cid:110) p n + M (cid:88) m =1 p nm · (cid:0) | c (0) nm, − | + 12 · | c (1) nm, − | (cid:1)(cid:111) , (8)where | c (0) nm,κλ | , | c (1) nm,κλ | are the coefficients describingthe evolution of the three-level system. Starting from |↑↑(cid:105) , these coefficients take the form c (0) nm,κλ = g ,nm,κλ cos ( g c ,nm,κλ t ) + g ,nm,κλ g ,nm,κλ c (1) nm,κλ = − i g ,nm,κλ sin ( g c ,nm,κλ t ) g c ,nm,κλ (9)according to Eq. 10 in [55] up to a phase factor (see Eq. 11in [55], see also Appendix B.1. in [56]). Here, g c ,nm,κλ = (cid:113) g ,nm,κλ + g ,nm,κλ , and the coefficients g ,nm,κλ = √
22 Ω nm,κλ (10) g ,nm,κλ = √
22 Ω n + κ m + λ,κλ (11)are the couplings between the states |↓↓(cid:105) ↔ √ ( |↑↓(cid:105) + |↓↑(cid:105) ) and the states √ ( |↑↓(cid:105) + |↓↑(cid:105) ) ↔ |↑↑(cid:105) , respectively.To include the phenomenological decay, we modify theequations above in an analogous way to the case of asingle ion to | c (0) nm,κλ | = 1 g ,nm,κλ (cid:2) g ,nm,κλ / g ,nm,κλ / · cos (2 g c ,nm,κλ t ) exp( − γt )+ 2 g ,nm,κλ g ,nm,κλ cos( g c ,nm,κλ t ) exp( − γt )+ g ,nm,κλ (cid:3) | c (1) nm,κλ | = g ,nm,κλ g ,nm,κλ (cid:2) / − / · cos (2 g c ,nm,κλ t ) exp( − γt ) (cid:3) , (12)1with the decay terms added to the oscillating terms ofthe expected population.Transport could potentially produce a coherent stateof motion when coherent excitation is not properly re-moved at the end of the transport [12]. Therefore, wealso analyze the data assuming a coherent distributionof motion and derive an average occupation number ¯ n for each measurement. In general, this yields a smaller ¯ n compared to the case where we assume a thermal distri-bution. For small ¯ n (¯ n ≈ . n derived from the twoanalyses have overlapping 1- σ confidence intervals, sincethe two distributions are very similar in this regime. Forlarge ¯ n (¯ n > n derived from the two analyses havenon-overlapping 3- σ confidence intervals. However, thereduced- χ from analysis using a coherent distributionshows a larger deviation from 1 compared to the num-ber derived using a thermal distribution, indicating thata thermal distribution is more appropriate here. As a conservative description for all regimes, we present theresults derived using a thermal distribution in the maintext. Mode rotation in junction