arXiv:1001.2714v1 [quant-ph] 15 Jan 2010 Superfast Cooling S. Machnes $^{1}$, M. B. Plenio2 , $^{3}$, B. Reznik$^{1}$, A. M. Steane$^{3}$, A. Retzker2 1 Department of Physics and Astronomy, Tel-Aviv University, Tel Aviv 69978, Israel 2 Institut f¨ ur Theoretische Physik, Universit¨ at Ulm, D-89069 Ulm, Germany 3 QOLS, The Blackett Laboratory, Imperial College London, Prince Consort Rd., SW7 2BW, UK and 4 Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, U.K. (Dated: November 1, 2018) Currently laser cooling schemes are fundamentally based on the weak coupling regime. This requirement sets the trap frequency as an upper bound to the cooling rate. In this work we present a numerical study that shows the feasibility of cooling in the strong coupling regime which then allows cooling rates that are faster than the trap frequency with state of the art experimental parameters. The scheme we present can work for trapped atoms or ions as well as mechanical oscillators. It can also cool medium size ions chains close to the ground state. Introduction — Laser cooling is the main tool that en- ables the exploration of low temperature phenomena in atomic physics, at nano and micro systems as well as the rapidly developing field of quantum technologies. Fol- lowing the original ideas of Doppler cooling[1, 2], laser cooling has taken a central role in the physics of cold atoms and the number of ideas and their sophistication is growing continuously. In the quest to propose cooling schemes that can cool to lower and lower temperatures at ever increasing rates the complexity and the efficiency of laser cooling has progressed a long way. oscillators [11, 12] for which the method can break the temperature limit imposed by the finite Q factor. The superfast cooling concept — Consider a trapped system which is coupled to a TLS. The system is in an almost-harmonic potential, with engineered coupling be- tween the external dof and the internal ones. For a stand- ing wave configuration when the ion or atom is at the node we obtain in leading order the following Hamilto- nian: Two level systems(TLS) admit the Doppler cooling limit. This limit can be overcome by the Sisyphus mech- anism either for free[3–5] or trapped particles[6]. The recoil limit can be broken by higher level systems by Ve- locity Selective Coherent Population(VSCP)[7] trapping or by Raman cooling[8] for free particles and in their ana- log for bound particles, Dark State cooling[9] and Raman side band cooling[10]. Cooling schemes for trapped parti- cles can be applied for mechanical oscillators in the setup described in [11, 12]. However, all such cooling schemes rely on a weak cou- pling between the internal degrees of freedom (dof) and the ion’s external motional state. This results in a clear separation of time scales between the two, allowing adia- batic elimination of the faster of the two dof-s. Unfortu- nately due to the weak coupling nature, the trapping fre- quency necessarily becomes the upper limit to the cooling rate and thus the achieved cooling rates are a few orders of magnitude lower than the trap frequency. We here present a fundamentally novel way of cool- ing a system of trapped particles, achieving rates that are faster than the trapping frequency, and which grows with the laser intensity with no fundamental upper limit. We demonstrate by means of numerics the proposed prin- ciple on a linear ion trap and show in detail how, using optimized sequences of coupling pulses, rapid cooling can be achieved. The proposed method is robust to fluctu- ations of the laser and may be adapted to a very wide range of systems, this can be achieved, for neutral atoms [8], trapped ions at low temperature [15], and mechanical We start by discussing the idealized case to gain in- A natural question to ask is whether we can cool, i.e. create the sideband-like term, in times which are faster than the trap frequency. A positive approach is by nu- merical optimization, but this requires an important in- put which is the initial ‘point’, i.e., a guess of the cooling cycle. One may try to initiate the optimization with the schemes valid for the weak coupling regime such as sideband cooling or the dark state cooling schemes and extend them to strong couplings. This approach proves highly inefficient. In our work we approach the prob- lem from the opposite direction: we choose as our initial guess a cooling scheme in the impulsive limit (infinitely fast), where we could reach the ground state infinitely fast if we neglect the higher order terms in the Hamilto- nian. Starting from this point we optimize numerically to adapt finite couplings attainable in the lab. where η is the Lamb-Dicke parameter, ν is the trapping frequency, Ω is the effective Rabi coupling between the two internal levels, a is the phononic annihilation oper- ators and σ$_{θ}$ is the pseudo spin operator in the direc- tion θ in the x, y plane. To cool the system, we would like to transfer energy from the motional to the internal dof., which can be dissipativly reinitialized via optical pumping. In other words, we wish to implement the red sideband-cooling operator, aσ $^{+}$+ a $^{†}$σ$^{−}$, [23–25] . In side- band cooling this term is generated from the ˆ Xσ$_{x}$ term after a rotating wave approximation (RWA), which re- quires a time which is longer than the trapping frequency, where ˆ X is the position operator. H/ ¯ h = δσ$_{z}$ + νa $^{†}$a + η Ω ( a † + a ) σ$_{θ}$, (1)