Efficient Sympathetic Motional Ground-State Cooling of a Molecular

Home , Laser cooling

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Mg – , 47 56 clrion ecular ∗ 47 n + ¯ – .I h olwn we following the In ]. , 50 osand ions ∼ 60 onurlatoms neutral to ] 48 0 .I usdside- pulsed In ]. . |↑i 001 , 61 oashort- a to ) |↑i ]. o single a for , 44 7 |↓i , 53 , (SBC), 9 ω .The ). 46 7 and ] and ] T , ,as ], as , 57 40 – – 2 motional state [57, 62] and outside the Lamb-Dicke (a) regime even exhibits points of vanishing coupling for cer- |ñ tain initial motional states, effectively disabling further G w cooling beyond these points (see Fig. 2). While this 0 regime has been studied theoretically [63–65], no experimental investigations are known to us. w Here, we demonstrate fast and robust sympathetic |¯ñ T ground state cooling of a 24MgH+ molecular ion along |n=0 ñ |n-1 ñ |n ñ |n+1 ñ one direction of motion using a 25Mg+ cooling ion. Start- ing with a Doppler-cooled single trapped 25Mg+ ion, we investigate in Sec. II a novel repumping scheme for pulsed (b) 3 4 Raman sideband cooling, in which the excited electronic 2 P F=1...4 state in the cooling cycle is effectively quenched to the 3/2 9.2 zGH electronic ground state via coupling to a short-lived excited state. We demonstrate that the requirements on meeting the optimum RSB pulse length are significantly 2 P1/2 F=3 relaxed for the new quasi-continuous scheme compared Doppler cooling to conventional pulsed Raman SBC. As a consequence s+ Raman + ~280nm of the large linewidth of the Doppler cooling transition s 25 + p in Mg , a significant amount of motional state popu- 2 P lation is trapped above the point of vanishing coupling 1/2 strength for RSBs. We employ second order RSB tran- F=2 |↑ ñ 2 sitions to sweep the population beyond this point and S1/2 1.789 zGH RF |↓ ñ demonstrate in Sec. IV that ground state cooling with F=3 |aux ñ m = 2 3 Lamb-Dicke factors as large as 0.45 and motional level F populations up to n ∼ 120 becomes possible by employ- th Figure 1. (Color online) Principle of resolved sideband ing RSBs up to 8 order. By optimizing the ground state 25 + st nd cooling and implementation in Mg . (a) Resolved cooling rate in terms of pulse lengths for 1 and 2 order SBC requires the linewidth Γ of the transition to be smaller RSB and the time spent on cooling with each sideband than the trap frequency ωT, so that individual transitions order, a total cooling time as short as 500 µs for cooling can be selectively driven. With the cooling laser tuned to the st a single ion from Doppler temperature (n¯ ∼ 10) to the 1 order RSB, one quantum of motion is removed from the motional ground state is demonstrated. The experimen- system in each absorption event. In the Lamb-Dicke regime, tal results are supported by numerical Master equation a change in motional state is suppressed upon decay back to 25 + simulations of the system (Appendix A). In Sec. III this the |↓i state. (b) Relevant level structure for SBC of a Mg 24 + 25 + ion. Doppler cooling and Raman transitions are performed by scheme is extended to cool a MgH / Mg ion crystal 2 2 to the ground state by interleaved cooling of both axial coupling the S1/2 and P3/2 states. Compared to a previous implementation of SBC [66], a second laser is added for re- modes. After optimizing the RSB pulse length for each 2 2 pumping the |auxi state via the S1/2 ↔ P1/2 transition. RF mode and the time spent on cooling each, a total cooling radiation couples the states |↑i and |auxi. time of 2.5 ms is achieved, resulting in a mean residual motional excitation of 0.06(3) and 0.03(3) for the in- and out-of-phase mode, respectively. 2 ↔ P3/2, F =4,mF =4 transition for 400 µs yields a n¯ ∼ 10 as the starting point for the ground state cooling

II. SBC OF A SINGLE ION sequence. Coherent electronic and motional state manipulation is implemented via Raman transitions, cou- 2 A. Experimental setup pling the |↓i and |↑i ≡| S1/2, F =2,mF =2i states. No change in the motional state correspond to carrier (CAR) The current work is based on previous results [66], transitions, while addition of phonons correspond to blue where we demonstrated a pulsed Raman sideband cooling sideband (BSB) transitions upon changing the electronic scheme using a single laser system for 25Mg+. In brief, state from |↓i to |↑i. The Raman beams are generated + 2 a 25Mg ion is trapped via isotope-selective photoioniza- via acousto-optic modulators from the P3/2 laser and 2 tion in a linear Paul trap with axial and radial motional are detuned with respect to the P3/2 state by around frequencies of 2.21 MHz and ∼ 5 MHz, respectively. The 9.2 GHz. The beams have π and σ+ polarization to maxi- relevant levels for SBC of a single 25Mg+ ion are shown in mize the coupling strength and are at right angle to each 2 Fig. 1(b). A frequency-quadrupled ﬁber laser ( P3/2 laser other with the diﬀerence wavevector aligned along the in the following) provides the radiation for Doppler cool- axial direction of the trap. This results in a Lamb-Dicke ing, coherent control, and state detection on the 25Mg+ parameter of η = 0.3 [66] for the axial mode of interest 2 ion. Doppler cooling on the | S1/2, F =3,mF =3i ≡ |↓i here. The limited detuning of the Raman laser beams 3

2 12 ) 1 0 P , F =3,m =3 state and by using RSB pulses (a) (b) 1/2 F Ω 10 with equal length. The experimental sequence starts with 0.8 nd Doppler cooling. For SBC, we ﬁrst apply a series of 2 8 st 0.6 order RSB pulses followed by a series of 1 order RSB 6 pulses of ﬁxed lengths. After each SBC pulse a short 0.4 + 4 (tr = 3 µs) optical repumping pulse using the σ beam

Population (%) 2 2 0.2 of the Raman laser tuned to resonance and the P1/2 . Rabi freq. (units of

ff laser is applied to clear out the population left in |auxi 0 E 0 0 20 40 60 0 20 40 60 and |↑i. In the experiment the actual elapsed time be- Trap levels Trap levels ′ tween two SBC pulses equals tr =5 µs caused by delays from the control electronics. In addition, we can use the 2 Figure 2. (Color online) Motional state population and P1/2 laser and the RF coupling between |auxi and |↑i effective Rabi frequencies. (a) Typical population distri- to implement a quench coupling during the Raman RSB bution over motional states after Doppler cooling, correspond- pulses. This quench coupling opens up an additional de- ing to n¯ ≈ 10. (b) Effective Rabi frequencies as a function of cay channel for the |↑i state back to the ground state the motional state calculated with a Lamb-Dicke parameter |↓i with a controllable decay rate, implementing a fusion η = 0.30 according to Eq. (1). The effective Rabi frequencies for different RSB orders have zero points at different motional between pulsed and continuous sideband cooling. After st nd rd excitations. solid/dashed/dotted: 1 /2 /3 order RSB. SBC, the population in the motionally excited states is probed by driving a RF π-pulse to transfer all population from |↓i to |↑i, followed by a stimulated Raman adiabatic st leads to off-resonant excitations, which result in dephas- passage (STIRAP) pulse on the 1 blue sideband (BSB) ing and population loss during coherent manipulation as [67]. This maps motionally excited states (n > 0) onto further discussed in Sec. II D. This effect limits the detec- the state |↓i, while the motional ground state population tion fidelity of the motional ground state population and (n =0) remains in the state |↑i[68]. the final n¯ detectable after cooling. It can be reduced We optimize the cooling rate for a fixed total cooling by employing a separate Raman laser system with larger time Tc by changing the amount of time spent on cool- nd st detuning. ing via 2 and 1 order RSBs, characterized by a time In the previously implemented SBC scheme [66] the scaling factor α. Depending on the pulse length of the st nd nd Rabi frequency Ωn′,n of the Raman transition between 1 (2 ) order RSB pulses tR1 (tR2), we apply NR2 2 st motional state |ni and |n′i was calculated according to order and NR1 1 order RSB pulses [57, 62] αTc NR2 = 2 ′ ′ tR2 −η n (1 − α)T 2 sn>! N = c , R1 t R1 where Ω0 is the CAR Rabi frequency, n< (n>) denotes ′ a Z the smaller (larger) of n,n , and Ln(x) are the gener- where the sign ⌊x⌋ = max{n ∈ ,n ≤ x} denotes the alized Laguerre polynomials. This allowed the deter- floor function. Short padding pulses are added to keep Tc mination of the π-time for complete population trans- fixed. The total time spent on ground state cooling dur- fer on the 1st and 2nd order RSBs (see Fig. 2). This ing the cooling sequence as shown in Fig. 3 is expressed way, RSB π-pulses were applied to sweep the popula- as tion starting from n = 40 to the motional ground state. ′ After each RSB pulse, multiple repump cycles involv- Ttotal = Tc + Nr · tr, (3) ing optical excitation of the |↑i → 2P transition and 3/2 where N is the total number of repump pulses. RF state inversion pulses between |auxi and |↑i are nec- r Since we are interested in the cooling time, we inves- essary to transfer the population back to the |↓i state, 2 tigate the dynamics of SBC by probing the population since the excited state in P3/2 can also decay into the 2 in the motional ground state with the STIRAP sideband |auxi ≡ | S1/2, F =3,mF =2i state. The cooling speed pulse for different SBC times Tc instead of optimizing for of this implementation was mainly limited by the radio lowest n¯. Assuming a constant cooling rate W during the frequency (RF) coupling strength and the required num- SBC cycle, the mean occupation of the motional states ber of sweeps (2-3) to reach the ground state. decays exponentially [69] as

−W t −W t n¯(t)=¯nf (1 − e )+¯nie (4) B. Cooling scheme

where n¯i and n¯f are the initial and ﬁnal mean occupa- In the approach described in the following, we signiﬁ- tion, respectively, and W is the cooling rate. Assuming cantly reduce the cooling time by adding a dedicated re- a thermal distribution with mean occupation of n¯(t) af- pump laser for faster repumping of the |auxi state via the ter SBC for a duration t, the population in the motional 4

140 140 NR2x NR1 x s) (a) s) (b) µ µ ( (

0 120 0 120 T T P1/2 laser .. P1/2 laser .. 100 100

80 80 RF RF 60 60 Cooling const. Cooling const. 101 102 101 102 2nd RSB σ-beam 1st RSB σ-beam Pulse length 2nd RSB (µs) Pulse length 1st RSB (µs) 140 1

s) (c) (d) µ

Time ( 0.8 0 120 T 0.6 T = 63.5 µs Figure 3. (Color online) Sequence for single ion sideband 100 0 25 + cooling. Sequence for SBC a single Mg . The sequence 0.4 nd st 80 starts by repeating NR2 2 order RSBs, followed by NR1 1 order RSBs with ﬁxed pulse lengths. After each SBC pulses, 0.2 2 60 Motional excitation repumping pulses consisting of P1/2 and σ beams are applied Cooling const. 0 to bring the ion back to |↓i state. 0 0.2 0.4 0.6 0.8 0 500 1000 Time scaling factor α Cooling time Tc (µs)

Figure 4. (Color online) Dependence of the cooling time ground state can be expressed as constants on experimental parameters. Cooling time nd constants as a function of the pulse lengths of the (a) 2 order 1 st P (t)= and (b) 1 order RSB pulses with α = 0.5 while fixing the 0 1+¯n (1 − e−W t)+¯n e−W t f i (5) non-scanned pulse length near its optimal value. blue point 1 (red square): Experimentally determined cooling constants = . −t/T0 with the quench coupling on (off) during the RSB pulse. The 1+¯nf +(¯ni − n¯f )e −1 quench coupling induces an effective decay rate of 42 ms from the |↑i state to the |↓i state. (c): Cooling time constant The desired cooling time constants T0 = 1/W for sets as a function of the time scaling factor with tR2 = 10 µs and of parameters (tR1, tR2, α) are extracted by fitting data tR1 = 10 µs. The lines in (a)-(c) are the result of Master to this model function in which only one parameter is equation simulations using the experimental parameters. (d) changed and n¯i and n¯f are common fit parameters. By Residual motional excitation as a function of the total cooling choosing Tc ∼ 7 · T0, we ensure reaching the steady state. time Tc. The line is the fit to the experimental data, which gives the cooling constant T0.

C. Optimization of experimental parameters lengths of around 10 µs. The experimentally determined Using the procedure described in the previous section, π-time for a RSB pulse starting at |n =1i is Tπ ≈ 16 µs. we vary the pulse length of the RSB pulses, the time scal- Averaging over the π-times for 1st and 2nd order RSBs ac- 2 ing factor α and the optical power of the P1/2 repump cording to Eq. (1), weighted by the thermal occupation of laser to optimize the cooling rate. Furthermore, we ex- the initial states, limited for each sideband to the states plore the transition between pulsed and quench cooling where the coupling strength of the respective sideband 2 by adding the P1/2 repump laser and RF coupling be- dominates, gives for both sidebands average π-times of tween the |auxi and |↑i states during the RSB pulses. ∼10 µs in agreement with the experimentally optimized A numerical simulation based on optical Bloch equations values. The fact that a constant π-time is sufficient for described in Appendix A supports our findings quantita- efficient cooling is a consequence of the relatively small tively. variation of the Rabi frequencies in the relevant range of motional levels as can be seen from Fig. 2. In standard pulsed Raman SBC, the cooling time con- 1. Pulse length optimization stant strongly rises upon deviation from the optimum pulse lengths [red points and dashed lines in Fig. 4(a) We first optimize the cooling time constant as a func- and (b)]. This dependence becomes much weaker for long tion of the pulse length for the 1st and 2nd order RSB pulses [blue points and solid lines in Fig. 4(a) and (b)] pulses. For each parameter set, a scan of the SBC time Tc by applying the quench coupling during the RSB pulses is performed, which allows us to derive the corresponding which opens a decay channel for the |↑i state. In the ex- cooling time constant as shown in Fig. 4(d). In Fig. 4(a) periment, this decay rate was adjusted to about 42 ms−1 and 4(b), the experimentally determined cooling time or a decay constant of about 24 µs to achieve the highest constants T0 are plotted against the pulse lengths. The cooling rate. At this decay rate, the effect of repumping st nd shortest T0 is obtained for 1 and 2 order RSB pulse is negligible for pulses shorter than the optimum pulse 5

1 rsb bsb (a) (b) (c) where ρ (t) and ρ (t) are the excitation probabilities st 0.8 on the 1 RSB and BSB at time t, respectively. For this analysis we subtracted a constant signal background from 0.6 the data. With the SBC sequence described in Sec. IIB, we achieve n¯ = ρrsb/(ρbsb − ρrsb) ≈ 0.01(2) as shown in 0.4 Excitation Fig. 5(a)-(c) after a cooling time of Tc = 500 µs. This 0.2 represents a reduction in cooling time by more than a factor of 30 compared to reference [66]. 0 -2.2 -2.1 -0.4 -0.2 0 0.2 0.4 2.1 2.2 2.3 Detuning from | ↓i ↔ | ↑i transition (MHz) III. SYMPATHETIC GROUND-STATE 25 + Figure 5. (Color online) Rabi excitation of Mg after COOLING OF A MOLECULAR ION sideband cooling. (a-c) Frequency scans of Raman transi- st st tions over 1 RSB (red), carrier (black) and 1 BSB (blue) transitions after SBC of a single 25Mg+. In this Section we adapt the quasi-continuous cooling scheme for sympathetic cooling of a molecular ion using 25Mg+ as the cooling ion species. length and the scheme is equivalent to standard pulsed SBC. For RSB pulses longer than the optimum pulse A. Loading and sympathetic Doppler cooling of length, the ion in the |↑i state decays back to the |↓i 24 + state through the new channel. The ion cycles between MgH the |↓i and the |↑i states and therefore more than one phonon can be removed within a single RSB pulse. The A two-ion crystal consisting of a 25Mg+ and a 24MgH+ scheme thus becomes equivalent to continuous SBC. This ion is prepared by isotope-selective photo-ionization load- quasi-continuous cooling scheme is insensitive to the ex- ing [71] of a pair of 25Mg+ and 24Mg+ ions with the act pulse length of the RSB pulses and provides high Doppler cooling laser tuned near the resonance of the robustness against intensity/pointing fluctuation of the 25Mg+ cooling transition. Then, the laser is tuned near Raman lasers. to the cooling resonance of 24Mg+ and hydrogen gas is leaked into the vacuum system through a leak valve, in- creasing the pressure up to ∼ 5 × 10−9 mbar. After a 2. Time scaling factor optimization photo-chemical reaction during which the excited 24Mg+ ion reacts with a hydrogen molecule to form 24MgH+ 24 + The time scaling factor α as defined in Eq. (2) dis- [72], the fluorescence of the Mg ion vanishes. After tributes the SBC time Tc into the time spent on the closing the leak valve and tuning the laser back to the nd st 25 + 2 order RSB TR2 = αTc and on the 1 order RSB Mg resonance, cooling commences and the mass of TR1 = (1 − α)Tc. Due to the dependence of the Rabi fre- the dark ion is determined via mass spectrometry using quency on the motional quantum number (see Fig. 2 and parametric heating of the two-ion crystal [73]. Since the Eq. (1)), the 2nd order RSB pulses are more efficient in mass of the two ions is almost identical, we do not expect cooling the population in motional states n > 20 at the significant deviations from Doppler cooling temperature, starting stage of the SBC cycle. For the lower motional even in the presence of additional heating [74]. states, the 1st order RSB pulses become more efficient, so that an α that is too large also increases the SBC cooling time constant (Fig. 4c). Since the variation of 1st and B. Two-mode sympathetic ground-state cooling 2nd order RSB Rabi frequencies with n are very slow, sequence the cooling time spent on either order only significantly influences the cooling time constant for the extreme cases For sympathetic ground state cooling of the molecu- of α → 0 and α → 1 and remains otherwise flat. lar ion we use a slightly modified pulse sequence compared to Sec. IIB. In contrast to the case of a single 25Mg+, the motion of the ions is described by two D. Cooling results modes along the axial direction, the in-phase (ip) and the out-of-phase (op) mode with secular frequencies of ip op The final n¯ after SBC is extracted from the red and ωT = 2π × 2.21 MHz and ωT = 2π × 3.85 MHz. The blue sideband excitation. Assuming that the ion stays in Lamb-Dicke parameters for the coupling of the Raman 25 + a thermal distribution after SBC, the ratio of excitations lasers to the Mg ion are ηip = 0.21 and ηop = 0.16. on 1st red and blue sideband fulfills the following relation With these Lamb-Dicke parameters, the effective Rabi [70] frequencies show no zero-crossings over the range of trap levels with significant population after Doppler cooling, ρrsb(t) n¯ so no higher order RSB pulses are necessary. However, Q := = , (6) ρbsb(t) 1+¯n with Lamb-Dicke factors as large as these, cooling of a 6

N resx Nopx 800 (a) 800 (b) s) µ ( 0 T P1/2 laser .. P1/2 laser .. P1/2 laser .. 600 600

400 400 RF RF RF

200 200 Cooling const. σ σ σ -beam -beam -beam RSB ip RSB op RSB ip 101 102 0 0.5 1 ′ Pulse length RSB (µs) Time scaling factor α Time Figure 7. (Color online) Dependence of the cooling time Figure 6. (Color online) Sequence for sympathetic side- constant on experimental parameters for sympathetic band cooling of a two-ion crystal. The in-phase and sideband cooling a two-ion crystal. (a) The cooling time out-of-phase axial modes of a 25Mg+/24MgH+ 2-ion crystal constant as a function of the optimal pulse length of the Ra- are cooled in an interleaved fashion to ensure simultaneous man RSB pulses on the ip (blue circles) and op (red squares) cooling. mode. (b) The cooling constant as a function of the optimal ′ time scaling factor α . The lines connect the points and are guides to the eye. single mode is not sufficient to enable high fidelity op- erations involving only this mode. The other mode acts as a spectator mode which modifies the effective Rabi (α′ = 0.5) and ensure near steady-state conditions by frequency of the mode of interest depending on its mo- performing cooling up to Tc ≈ 2.5 ms. The highest cool- tional state [62]. Therefore, we have implemented an ing rate is observed with a pulse length of 15 µs (20 µs) interleaved pulse sequence for SBC both axial modes si- for the RSB on the ip (op) mode as shown in Fig. 7a. (ip) (op) multaneously as shown in Fig. 6. Depending on the time With a π-time of Tπ ≈ 21 µs and Tπ ≈ 26.5 µs for ′ scaling factor α , which in this case distributes the total the 1st RSB starting in the |n =1i state, the Rabi fre- SBC time Tc into time spent for RSB on the ip mode quency according to Eq. (1) averaged over the thermal ′ ′ (1 − α )Tc and time for RSB on the op mode α Tc, we occupation of motional states results in a mean π-time apply of 14 µs and 25 µs for the ip and op mode, respectively. (1 − α′)T As in the single-ion case discussed in Sec. IIC, the N = c (7) increase in the cooling time constant for pulse lengths ip t ip longer than the optimum pulse length is rather mild as a ′ α Tc consequence of the quench coupling during RSB pulses. N = (8) op t op pulses on the ip and the op mode, respectively. Here D. Time scaling factor optimization tip, top are the pulse lengths of the RSB pulses on the ip and the op mode. In the case Nip > Nop (Nip < Nop) The time scaling factor in the SBC sequence for cool- we start the SBC cycle with Nres = |Nip − Nop| pulses ing a two-ion crystal divides the total cooling time Tc into on the ip (op) mode, followed by Nop (Nip) pulses on the ip and the op mode in an interleaved fashion. After the time spent on cooling the ip mode and the op mode every single SBC pulse, a repumping pulse as described for the optimized pulse lengths derived from the mea- above is applied to clear out the |↑i and |auxi states in surements in the previous section. The measured SBC cooling time constants for the two modes as a function of addition to the quench coupling present also during the ′ ′ RSB pulses. Optimizations similar to the case of a single α are shown in Fig. 7b. Although an α that is too small 25Mg+ are performed to minimize the total duration of (large) accelerates the cooling of the ip (op) mode, the SBC. cooling time of the opposite mode increases. The optimal value for the time scaling factor α′ ≈ 0.5 is determined by the point where SBC of both the ip and op mode are C. Pulse length optimization achieved with the shortest total duration of the SBC cycle. Interestingly, this corresponds to a different number of RSB pulses for the two modes. The length of the RSB pulse on the ip (op) mode is optimized by fixing the other mode’s RSB pulse length. As described in Sec. IIB, the cooling time constant is derived from the motional ground state population mea- E. Cooling results sured by performing a STIRAP pulse resonant with the st 1 BSB of either mode for different cooling times Tc. We Fig. 8 shows frequency scans of the carrier and both spent an equal amount of time for cooling on both modes RSB and BSB transitions after SBC the ip and op modes 7

1 (a) (b) (c) (d) (e)

0.5 Excitation

0 -3.8 -3.75 -2.2 -2.1 -0.2 0 0.2 2.15 2.2 2.25 3.75 3.8 3.85 Detuning from | ↓i ↔ | ↑i transition (MHz)

Figure 8. (Color online) Rabi excitation of 25Mg+ after 2-mode SBC a 24MgH+/25Mg+ crystal. Frequency scans of st st Raman transitions over the 1 RSB (red) of the (a) op and (b) ip modes, (c) the carrier (black), and the 1 BSB (blue) (d) op and (d) op modes after SBC both modes on 25Mg+. along the axial direction with optimized parameters. The band orders and apply to all of them the same pulse final n¯ is determined from the red and blue sideband length, thus extending the sequence shown in Fig. 3 to excitations of each mode as described in Sec. II D. As higher-order modes. This is a valid approach, since the before, the offset from off-resonant Raman excitations variation in maximum Rabi frequency across the higher has been subtracted for the analysis. We reach a n¯ip = order sidebands is small. Simultaneously to each RSB 0.06(3) for the ip mode and n¯op =0.03(3) for the op mode pulse, we apply the quench coupling to further reduce after a total cooling time Ttotal ∼ 2.5 ms. Compared to the sensitivity to the optimum pulse length. After each an extension of the SBC scheme described in reference RSB pulse, a repumping pulse as described before is ap- [66] to 2-mode ground state cooling, the optimized new plied. The optimal number of applied sideband orders, scheme reduces the cooling time by a factor of 8. derived from measurements of the cooling time constant as a function of the maximally applied sideband order, is shown in Fig. 10a. In the optimal sequence, we need to th IV. SIDEBAND COOLING BEYOND THE apply sideband pulses up to the 8 order, which confirms LAMB-DICKE REGIME the prediction of Fig. 9b. More generally, with known Lamb-Dicke parame- For some applications large Lamb-Dicke factors are deter and thermal distribution over motional states after sirable, since they enhance the sensitivity of the ion’s Doppler cooling, a corresponding SBC strategy can be motion to small forces. This is the case for photon used to achieve the most efficient cooling as illustrated recoil spectroscopy [11] or the non-destructive internal in Fig. 10b, where we define the cooling efficiency as state detection of a molecular ion using oscillating dipole β(n) · Ωn−β,n. For a selected upper trap level to be ad- forces [75–77]. In the following we extend the single- dressed by cooling and a known Lamb-Dicke parameter, ion ground state cooling scheme presented in Sec. II to enable efficient ground state cooling outside the Lamb-

5 ) 1 (a) 0 (b)

Dicke regime, which is readily extended to the multi-ion Ω case discussed Sec. III. 4 0.8 At a trap frequency of 1 MHz and a temperature of 3 0.6 1. 1 mK theoretically achievable with Doppler cooling a 2. 3. 25 + 4. 5. Mg ion, states up to n ∼ 120 are signiﬁcantly popu- 2 0.4 6. 7. 8. lated, leaving less than 0.3% population in levels n> 120 Population (%) 1 0.2

(Fig. 9a). At this trap frequency, the Lamb-Dicke param- . rabi freq. (units of ff eter in our system is η = 0.45. The effective Rabi fre- 0 E 0 0 50 100 0 50 100 quency for RSB transitions depends strongly on the trap Trap levels Trap levels levels and shows several points of vanishing coupling over the range of trap levels with significant population (see Figure 9. (Color online) Motional state population and Fig. 9(b) and Eq. (1)). Employing pulses on the 2nd or- 25 + effective Rabi frequency for small trap frequency. (a) der sideband as in the scheme for a single Mg is no Distribution of the motional levels at a motional frequency of longer sufficient. Instead, we employ as many higher or- ωT = 1 MHz and a temperature of T = 1mK corresponding der sidebands as necessary. For larger motional states, to the Doppler cooling limit of the 25Mg+ ion. Less than successively higher order sidebands exhibit a maximum 0.3% of the population are in the trap levels |n> 120i. (b) in their coupling rate. Moreover, pulses on higher order Effective Rabi frequencies at η = 0.45 for different RSB orders sidebands are more efficient since more than one phonon are shown as a function of the trap levels. For ions located in is removed per sideband pulse. For simplicity, we split higher trap levels, RSB pulses of higher orders β(n) are more efficient for cooling. the total cooling time Tc equally between all the side- 8

0.6 1 forces through the excitation of the ion’s motion from (a) (b) s)

µ the ground state, such as in photon recoil spectroscopy 500 0.5 3 [11] and the detection of electric ﬁelds [61, 78]. 0.4 400 5 The demonstrated motional ground state cooling of a η 0.3 molecular ion, in particular when combined with cooling 300 7 at low trap frequency, represents an important step to- 0.2 wards the implementation of non-destructive state prepa-

Cooling constant ( 9 200 0.1 2 4 6 8 10 20 40 60 80 100120 ration and detection techniques [75–77] based on the de- Applied SB orders Trap levels tection of small optically-induced and state-selective dis- placement forces acting on a trapped molecular ion. Figure 10. (Color online) Determination of the maximum sideband order. (a) The cooling time constant as a function of applied sideband orders. The optimal number for ACKNOWLEDGEMENTS the sideband orders determined experimentally confirms the prediction shown in Fig. 9(b). (b) RSB orders with highest We acknowledge the support of DFG through QUEST effective cooling rate as a function of the trap level and the Lamb-Dicke parameter. With known Lamb-Dicke parameter and grant SCHM2678/3-1. Y.W. acknowledges support and motional distribution a corresponding SBC sequence can from IGSM. We thank R. Blatt for generous loan of be adopted. equipment. the sideband order with highest cooling efficiency is plot- Appendix A: Numerical simulation for SBC ted. This strategy can be combined with multi-mode cooling through interleaved pulse sequences as shown in The dynamics of the system during quasi-continuous Sec. III. SBC is modelled using optical Bloch equations, where two electronic states |↑i and |↓i and 80 trap levels (Fig. 11) are considered. The Raman RSB pulses V. SUMMARY & DISCUSSION are included as a resonant coupling between |↓,ni and |↑,n − βi with β indicating the sideband order used. The A quasi-continuous sideband cooling scheme was in- spontaneous decay during the Raman cooling cycle via troduced by applying a quench coupling to the excited the auxiliary states [80] is implemented with an effec- state in pulsed Raman sideband cooling. For long Raman tive decay rate Γeff determined experimentally. This ap- pulses this scheme allows multiple RSB–spontaneous proach neglects effects from the actual multi-level elec- emission cycles, reminiscent of continuous quench cool- tronic structure of 25Mg+. Repumping after applica- ing. This feature makes SBC more robust and signif- tion of the RSB is implemented through a projection icantly reduces the optimization required for optimum of the population onto the electronic ground state |↓i. cooling, since RSB pulses that are longer than the π-time for a specific initial motional state contribute more effi- ciently to cooling. Using this scheme, we performed an optimization of the cooling rate of single-ion ground state cooling and two-ion sympathetic ground state cooling, significantly outperforming previous results [66]. These findings are important to minimize the time required for ground state cooling, thus reducing overhead in experiments with trapped ions. The results are applicable to other commonly used hyperfine qubit systems such as 9Be+, 111Cd+, or 171Yb+. A variation of the scheme can be applied to other systems like 40Ca+, 24Mg+ in high magnetic field or optically trapped neutral atoms. We have also demonstrated experimentally that SBC Figure 11. (Color online) Relevant levels for numerical to the ground state is possible outside the Lamb-Dicke simulation of the dynamics during the Raman SBC regime by employing higher-order RSB transitions, thus cycles. The Raman lasers are tuned to be resonant with confirming theoretical predictions [63–65]. This regime the βth order RSB (β = 1 in graph). Through the dissi- is particularly relevant for systems where the Doppler pative channels either from off-resonant excitation or via a cooling linewidth significantly exceeds the motional fre- repumping process followed by spontaneous emission, the ion −1 quency as is the case for optically trapped neutral atoms, is reinitialized to the |↓i with a decay rate of 0.005 s and −1 or in situations where the Lamb-Dicke factor is delib- 0.47 s , respectively. Spontaneous emission happens on the erately chosen as large as possible. This is the case carrier transition with a probability 1 − ξ and the remaining for example in experiments which aim to detect small small fraction ξ happens on the RSB transitions. 9

The spatially averaged Lamb-Dicke factor for sponta- and Ωn,n−β as the effective Rabi frequency coupling the neous emission along the axial direction is η˜ = 0.134. two states |↓,ni and |↑,n − βi. This system of differential Heating from spontaneous emission is taken into account equations is solved numerically to produce the evolution through a branching ratio of 1 − ξ : ξ for emission on during a single Raman cooling pulse. the CAR and RSB transition. Here ξ is on the order of 3 · η2 ≈ 0.05, where the factor 3 considers the multi- For every set of parameters (T , α, t and t ), the ple scattering events until the ion falls into the |↓i state. c R1 R2 evolution during the pulse sequence as shown in Fig. 3 Combining all the ingredients above, we end up with the e is computed by repeatedly solving the equations above. following optical Bloch equations The actual evolution time for the atomic system dur- Ω ing each pulse is reduced by 1 µs from the chosen pulse ρ˙ = i n−β,n ρ − ρ ↓n,↓n 2 ↓n,↑(n−β) ↑(n−β),↓n lengths to include realistic experimental pulse area reduction. The final population in the motional ground + (1 − ξ) · Γ ρ eff ↑n,↑n state is considered as the signal detected by the STIRAP + ξ · Γeff ρ↑(n−1),↑(n−1) (A1) pulse and corrected by an amplitude a =0.7 and an off- ρ˙↓n,↑(n−β) = − ρ˙↑(n−β),↓n set b =0.23 reduction due to experimental imperfections Ω according to = i n−β,n ρ − ρ 2 ↓n,↓n ↑(n−β),↑(n−β) Γeff − ρ (A2) 2 ↓n,↑(n−β) Ωn−β,n ρ˙ = i ρ − ρ y = a × (ytheory − b). (A5) ↑(n−β),↑(n−β) 2 ↑(n−β),↓n gn,↑(n−β) − Γeff ρ↑(n −β),↑(n−β) (A3) with the density matrix elements defined in the usual way, e.g. A scan over the SBC cooling time Tc reproduces the experimental data, which are processed in a similar way as ρ↓n,↑(n−β) = h↓,n| ρ |↑, (n − β)i (A4) described in Sec. IIB.

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