3 FIG. 2. Repeated applications of the sample cycles. Inset: X and P pulse lengths for sample cooling cycle C. Neg- ative pulse lengths indicate the spin is to be oriented in the negative x or y directions. Each block of 6 bars (3 X and 3 P pulses) represent a single cooling sequence. These sequences are interspaced by reinitialization of the ion’s internal state to make up a complete cooling cycle. TABLE I. Cooling cycle performance. We define a cooling sequence as a series of alternating ˆ Xσ$_{x}$ -pulses and ˆ P σ$_{y}$ -demi- pulses of varying lengths, followed by a reinitialization of the ion’s internal dof. A cooling cycle is comprised of several cooling sequences, which are generally non-identical. FIG. 1. Initial and final energy (above ground, in units of ¯ hν ) for a single application of the cooling cycle and for 25 applications. Cycle A Cycle B Cycle C Initial energy [¯ hν ] 3 5 7 Final energy [¯ hν ] 0.4 1.27 1.95 Final energy after 25 cycles 0.02 0.10 0.22 Cycle duration 4.4 2 π v 2.7 2 π v 0.8 2 π v No. pusles per cycle 180 90 90 No. of sequences 10 10 10 0 1 2 3 4 5 6 7 8 0.5 1 1.5 2 2.5 3 3.5 Initial energy [phonons] Final energy [phonons] Cycle A Cycle B Cycle C No cooling 0 2 4 6 8 10 12 0.5 1 1.5 2 Initial energy [phonons] Final energy [phonons] Cycle A Cycle B Cycle C No cooling 0 5 10 15 1 2 3 4 5 6 7 Number of applications Final energy [phonons] Cycle A Cycle B Cycle C limit ( η Ω ≫ ν ), and applied for realistic parameter val- ues, where we may not neglect free evolution while puls- ing. The sequences presented are just examples of what is achievable, and by no means are they to be considered canonical in any way. All computations below are done for the following physical settings, which are achievable in the lab: $^{40}$Ca$^{+}$, ν = 1 MHz · 2 π , Ω = 100 MHz · 2 π , λ$_{laser}$ = 730 nm , giv- ing a Lamb-Dicke value of η = 0 . 31, [14]. Table 1 details 3 sample cycles and their cooling performance. Note that the coupling terms in the propagator are proportional to t$_{p}$η √ 2 mν ¯ h Ω ˆ Xσ$_{θ}$ and ηt$_{f}$ t$_{p}$ Ω ν ˆ P σ$_{y}$ and therefore total cooling time scales as 1 $_{/}$√$_{Ω since}$ t$_{p}$ × t$_{f}$ scales like 1 / Ω. We made the optimization in the impulsive limit, and then applied the deduced sequence using the full Hamil- tonian. The results are shown in Fig. 1. Starting from a thermal state of varying temperatures, Fig 1a shows the final energy for a single application of a cooling cycle; Fig 1b shows the result of 25 applications of the same cycle. The latter cannot be deduced directly from Fig 1a because the state is no longer thermal after one or more cooling cycles. Note how the cycles give a good cool- ing for a wide range of initial temperatures, even though optimization was carried out for a specific initial temper- ature In Fig. 2 we show how repeated applications of the three sample cycles continues to lower the energy, until a steady state, specific to each sequence is achieved. An example timing of sequences within a cycle is shown in fig. 2 inset. Robustness — The robustness of the cooling pulses is extremely important for experimental realization. In or- der to analyze the robustness we have simulated the op- eration of the pulses under noisy conditions of the lasers. The results indicate that the cooling pulses are extremely robust even though they were not optimized to be so. In Fig. 3 we study the robustness of cycle C, repeated 25 times, to Gaussian noise in the pulse timings. We assume that instead of the prescribed pulse time t , we implement t −→ (1 + ϵ ) t , with ϵ being drawn from a Gaussian dis- tribution with a varying standard deviation. A similar noise pattern was applied to the laser power, Ω. The superfast cooling exhibits two slightly different sensitiv- ities to noise: for the short time-frame ( ≪ ν − $^{1}$), which interferes with the commutation relations allowing the creation of the effective P pulse, sensitivity is somewhat higher, while for noise occurring on longer timescales, i.e. the duration of the various pulses comprising the cooling sequence, Superfast cooling is extremely robust. The plots show the mean final phonon count over 500 cy- cles. A graceful degradation in cooling performance can be observed. Note that the state of the system after cooling is not a thermal state. This is due partly to purely numeri- cal issues stemming from the need to exponentiate what are essentially infinite-sized Hamiltonians, and partly a physical feature of the specific cooling sequences pre- sented. The latter results from the necessity to make use of relatively small density matrices in initial optimization stages, which resulted in effective cooling operators that do not cool high harmonic modes (above 30) as well as