content_image/423921.jpg content_image/423922.jpg
<img src='content_image/423921.jpg'>

they cool lower modes. While this difference is negligible before cooling, when going significantly below 1 phonon, the remaining energy in the high modes becomes a not- insignificant part of the overall energy. We believe that with higher computational power sequences resulting is more thermal-like final states are achievable. 

Ion Chain —- By applying this cooling method to a large chain a product state of ground states can be reached, i.e., we will cool to the | 0 〉 | 0 〉 .... | 0 〉 state in the local basis. This state is not far from the global ground state. In fig. 4 we show the number of phonons in the center of mass mode for a regular ion trap and for a trap in which all the ions are equidistantly pinned. In the latter case a saturation value of around 0.2 phonons is reached for the center of mass mode while for the regular trap there is no staturation but the population is still low for moderate chains. 

Standing wave —- There are a few ways to gener- ate the standing wave Hamiltonian. We are not re- ally interested in a standing wave but in a pulse of a standing wave. We can emulate this pulse by run- ning wave pulses by using the two following pulses: i Ω ( σ$_{x}$ + ηx ) , − i Ω ( σ$_{x}$ − ηx ). These pulses can be cre- ated by passing a pulse through a beam splitter and bringing it through two different directions, thus con- trolling the phase and sign of the Lamb Dicke param- eters independently. The unitary transformation which describes this is: e i Ω $^{σ$_{x}$}$eiη Ω σ$_{x}$$^{x}$e − i Ω $^{σ$_{x}$}$eiη Ω σ$_{x}$x = e i 2 η Ω σ$_{x}$$^{x}$, which creates the unitary operation that we need. More- over, since we are interested in fast cooling rates we also are ensured that the standing wave will not drift. The creation of the standing wave can be done as in[26]. In the case of mechanical oscillators this trick would not 

<img src='content_image/423922.jpg'>

work since there is no temporal control of the effective Lamb Dicke parameter, but if the coupling is with Ra- man transitions a different intermediate level can be used and thus the effective Lamb Dicke will be different while the Rabi frequency may be the same and thus cancel the first pulse. 

So far we have assumed the ion can be modeled by a non-dissipative two-level system with effective Rabi fre- quency Ω. This is a simplification. Some experimental setups use a 3-level system with the effective Rabi cycle achieved by way of a highly detuned Raman transition off a highly-dissipative level [23–25]. As a result, the true laser coupling of 100 GHz is reduced to an effective co- herent 10 MHz transition. However, unlike quantum in- formation processing, superfast cooling does not require a fully coherent transition. We could reduce the detuning and achieve a higher Rabi frequency. At Ω = 1 GHz , for example, the cooling process speeds up, requiring only 0 . 4, 0 . 3 and 0 . 2 2 π ν for cycles A, B and C, respectively. 

Nanomechanics — This scheme may have increasing importance in the nonmechanical and hydromechanical world since there the finite Q value limits the final tem- perature. One would use the setup described in [11, 12]; i.e. couple a TLS to the oscillator. In this case the ef- fective Lamb Dicke parameter is on the order of η = 0 . 03([12]) and can be much higher for the scheme which is generated with magnetic gradients([11]). In both of these schemes the strong coupling regime can be reached by strong lasers or even microwaves. The method cur- rently in use [27, 28], assisted by a cavity, cannot beat the limit of the trap frequency due to the Gaussian na- ture of the scheme. 

Conclusion— To conclude, we have introduced the optimized-control approach to cooling trapped ions, by 

