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Physical Review A

Cooling of a trapped ion in the strong-sideband regime

C. A. Blockley, Daniel F. Walls

Published Mar 1, 1993• 40 citations
Publisher

Abstract

A single two-level atom, with quantized center-of-mass motion, is constrained to move in a one-dimensional harmonic potential while interacting with a single-mode classical traveling-wave light field. When the classical light field is tuned to the atom's lower vibrational sideband, cooling can occur. The strong-sideband and Lamb-Dicke perturbation regimes for the system are defined. The steady-state and time-evolution behaviors in the strong-sideband regime are discussed and, in particular, it is shown that the steady-state average trap number saturates when spontaneous emission is weak and that the steady-state average trap number depends more strongly on the trap frequency in the saturated regime than previously predicted. Finally, the possibility of observing quantum jumps between trap levels is discussed.

DOI
10.1103/physreva.47.2115

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A single two-level atom, with quantized center-of-mass motion, is constrained to move in a one-dimensional harmonic potential while interacting with a single-mode classical traveling-wave light field. When the classical light field is tuned to the atom's lower vibrational sideband, cooling can occur. The strong-sideband and Lamb-Dicke perturbation regimes for the system are defined. The steady-state and time-evolution behaviors in the strong-sideband regime are discussed and, in particular, it is shown that the steady-state average trap number saturates when spontaneous emission is weak and that the steady-state average trap number depends more strongly on the trap frequency in the saturated regime than previously predicted. Finally, the possibility of observing quantum jumps between trap levels is discussed.
Abstract Trapped ion in the Lamb–Dicke (LD) regime with the LD parameter η ≪ 1 can be cooled down to its motional ground state using sideband cooling. Standard sideband cooling works in the weak sideband coupling (WSC) limit, where the sideband coupling strength is small compared to the natural linewidth γ of the internal excited state, with a cooling rate much less than γ. Here we consider cooling schemes in the strong sideband coupling (SSC) regime, where the sideband coupling strength is comparable or even greater than γ. We derive analytic expressions for the cooling rate and the average occupation of the motional steady state in this regime, based on which we show that one can reach a cooling rate which is proportional to γ, while at the same time the steady state occupation increases by a correction term proportional to η 2 compared to the WSC limit. We demonstrate with numerical simulations that our analytic expressions faithfully recover the exact dynamics in the SSC regime.
We study laser cooling of two ions that are trapped in a harmonic potential and interact by Coulomb repulsion. Sideband cooling in the Lamb-Dicke regime is shown to work analogously to sideband cooling of a single ion. Outside the Lamb-Dicke regime, the incommensurable frequencies of the two vibrational modes result in a quasicontinuous energy spectrum that significantly alters the cooling dynamics. The cooling time decreases nonlinearly with the linewidth of the cooling transition, and the effect of dark states which may slow down the cooling is considerably reduced. We show that cooling to the ground state is also possible outside the Lamb-Dicke regime. We develop the model and use quantum Monte Carlo calculations for specific examples. We show that a rate equation treatment is a good approximation in all cases.
The dynamics of a two-level atom driven by a single laser beam and three-level atom (Lambda configuration) irradiated by two laser beams are studied taking into account of the quantized center-of-mass motion of the atom. It is shown that the trapped atom system under appropriate resonance condition exhibits the large time-scale revivals when the index of the vibrational sideband responsible for the atomic electronic transition is greater than unity. The revival times are shown to be dependent on the initial number of vibrational excitations and the magnitude of the Lamb-Dicke parameter. The sub-Poissonian statistics in vibrational quantum number is observed at certain time intervals. The minimum time of interaction for which the squeezed states of motional quadrature are generated is found to be decreasing with the increase in the Lamb-Dicke parameter.
We present the results of simulations of optical sideband cooling of atomic ions in a trap with a shallow potential well. In such traps, an ion cannot be Doppler cooled near to the Lamb-Dicke regime $\mathrm{[}{\ensuremath{\eta}}^{2}(2\ensuremath{\langle}n\ensuremath{\rangle}+1)\ensuremath{\ll}1]$. Outside the Lamb-Dicke regime, the sideband cooling dynamics are altered by the existence of various Fock states with weak coupling where the cooling becomes very slow. A 40Ca+^{40}\mathrm{Ca}^{+}40Ca+ ion trapped in our Penning trap realizes such a situation; hence single stage cooling is inefficient to prepare the ion in the motional ground state. For these systems, it is necessary to study the cooling dynamics in detail and we show that it is possible to implement an optimized cooling sequence to achieve efficient ground-state cooling. We also present the simulated cooling dynamics of two ions trapped in a Penning trap, where the presence of an additional motional mode requires a complicated cooling sequence in order to cool both axial modes to the ground state simultaneously. Additionally, we demonstrate the dissipative preparation of Fock states outside the Lamb-Dicke regime by sideband heating a single ion in a Penning trap.
Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.