COHERENT COOLING OF ATOMS IN A FREQUENCY-MODULATED STANDING LASER WAVE: WAVE FUNCTION AND STOCHASTIC TRAJECTORY APPROACHES
V. Yu. Argonov*
Pacific O ceanological Institute, Russian Academy of Sciences 690041, Vladivostok, Russia
Received May 11, 2014
The wave function of a moderately cold atom in a stationary near-resonant standing light wave delocalizes very fast due to wave packet splitting. However, we show that frequency modulation of the field can suppress packet splitting for some atoms whose specific velocities are in a narrow range. These atoms remain localized in a small space for a long time. We demonstrate and explain this effect numerically and analytically. We also demonstrate that the modulated field can not only trap but also cool the atoms. We perform a numerical experiment with a large atomic ensemble having wide initial velocity and energy distributions. During the experiment, most of atoms leave the wave while the trapped atoms have a narrow energy distribution.
DOI: 10.7868/S0044451014110030
1. INTRODUCTION
Laser cooling and trapping of atoms is a rapidly developing field of modern physics. Cold particles in a laser field are a common physical substrate used in numerous fundamental and applied issues such as Bose Einstein condensates, quantum chaos, singleatom laser, quantum computer, etc.
In general, the idea of mechanical action of light on matter is rather old. As far as we know, it was first suggested by Kepler fl] in 1619 in order to explain a deviation of the comet tails nearby the Sun. In 1873, Maxwell first estimated the light pressure [2], and in 1899, Lebedev first measured it in experiment [3] with a macroscopic body. In the first half of the 20th century, analogous experiments with microscopic particles were carried out by Gerlach and Stern [4], by Ivapitza and Dirac [5], and by Frisch [6].
The modern paradigm of mechanical manipulation of atomic motion by the laser began to emerge in the second half of the 20th century. The discovery of a gradient dipole force acting 011 neutral atoms in an intensive variable field by Gaponov, Miller, and Askaryan [7, 8] was a theoretical basis for further results. In 1968, Letokhov theoretically predicted the trapping of atoms in the nodes or ant modes of a standE-mail: argonovfflpoi. dvo.ru
ing wave [9]. Soon, in the 1970s, first experimental methods of laser acceleration [10, 11] and cooling (the Doppler cooling) [12 15] of atoms were proposed. The basic theory of dissipative atomic motion in a laser field was built by Ivazantsev [16]. The theory considered the field in terms of the optical friction force acting 011 a moving atom. The friction force can be positive (atoms decelerate) or negative (atoms accelerate), and it 11011-linearly depends 011 the atomic velocity.
I11 1978, the Doppler cooling was first demonstrated in the experiment by Wincland and his collegaues [17]. I11 the 1980s, a series of other mechanical effects (predicted by early theoretical works) were also demonstrated experimentally: atomic nionochromatization in the velocity space [18], collimation of an atomic beam [19], beam diffraction in a standing wave, beam reflection from a wave ("laser mirror"), and channeling of atoms [20]. New methods of atomic cooling in a laser field were proposed: the Sisyphus cooling [21] and the velocity selective coherent population trapping (VSCPT) [22]. Experimental realization of various cooling techniques in the 1980 1990s established a series of temperature records. While the early 1980s experiments provided the temperatures of the order of 0.1 Iv [23], in 1990s, Nobel laureates Cliu, Cohcn-Tannooji, and Phillips reached the temperatures of the order of 0.2 fiK [24], and nowadays, sophisticated methods provide temperatures of the order of 0.2 11K [25].
I11 the 1990s, numerous new mechanical effects were
discovered in the study of cold atoms in a nonstation-ary field with modulation and jumps. The groups of Raizeii and Zoller [26 28] reported various effects related to dynamical chaos and the quantum classical correspondence (having not only a pure physical but also methodological importance). In particular, they first experimentally demonstrated some manifest at ions of chaos in quantum systems and measured the difference between predictions of semiclassical models and real quantum behavior (in the study of so-called dynamical localization). In the framework of our study, it is important to note the special possibility of atomic localization in phase space: atoms with special values of the initial positions and momenta can be trapped in the resonance stability "islands" embedded in a chaotic "sea" [28] (in terms of the dynamical systems theory [29, 30]).
Although the basic theory of atomic motion in a laser field was formed in the 1970 1990s, it contained a number of approximations and considered a limited class of physical systems. In most of studies, atoms were treated either as plane waves in the coordinate space (approximation valid when the spatial extent of the atomic wave packet is substantially larger than the wavelength of the field) or as dot-like particles (approximation valid when the atomic velocity is sufficiently-large). In recent years, the growth of computational power has provided tools for precise analysis of atomic motion beyond most of old approximations. Today, it is possible to model fully quantized atomic motion in terms of the wave function (atomic wave packets) or the density matrix. This helps study the regimes of small atomic momenta (of the order of the photon momentum), weak fields (of the order of few photons), small atom field dctunings (when intense Rabi oscillations occur and both resonant and nonrcsonant potentials [16] virtually coexist in a system), etc. In the quantum consideration, even comparatively simple systems (a standing wave or a two-level atom) demonstrate new effects (unknown in previous studies). For example, in [31], the splitting of traveling atomic wave packets on standing-wave nodes was discovered, and in [32], the anomalous atomic spatial concentration in the field (not fitting old semiclassical predictions) was demonstrated. In particular, it was shown that for some values of the field intensity, atoms can concentrate not only in the wave nodes or antinodes but also in intermediate positions. None of these effects could be demonstrated without precise quantum description of atomic motion (taking the mechanical photon recoil and finite atomic spatial and momentum unccrtainity into account).
In our studies, we focus on the quantized atomic dynamics in the regime of small atom field detuning. When an atom moves in a near-resonant standing light wave, two periodic optical potentials form in space [16]. When the atom crosses a standing wave node, it can undergo the Landau Zciier (LZ) transition between these two potentials. Such transitions cause splitting of atomic wave packets [31, 33] and rapid derealization of the wave function [34]. However, under some additional conditions, manifest at ions of atomic localization also appear. In [35], we reported that in a stationary field, the interference between packet splitting products can break the symmetry of LZ transitions and cause localization of atoms in the momentum space. In this paper, we study a similar quantum system, but in a modulated field. We show that frequency modulation of the field can suppress the splitting of wave packets for atoms having velocities in a specific narrow range (determined by the field modulation parameters). These atoms stay trapped in the field for a long time (the effect of velocity-selective trapping of atoms). We provide additional simulations showing that in an experiment, this effect may significantly decrease the energy distribution of moderately cold atoms, and can therefore be used for coherent laser cooling.
In this paper, we pay much attention to methodological aspects of the study. The paper provides three different approaches to the analysis of atomic motion. First, we demonstrate the manifest at ions of the velocity-selective trapping numerically by solving quantum equations (describing the dynamics of atomic wave functions). Second, we explain the effect theoretically using semiclassical model (describing the dynamics of dot-like atoms with continuous trajectories). Third, we develop a stochastic-trajectory model (similar to the hybrid model used in [31], describing the dynamics of dot-like atoms with piecewise continuous trajectories accompanied by occasional quantum jumps) and use it in a numerical experiment demonstrating the cooling of large atomic ensemble. We also provide additional numerical experiments demonstrating the similarity of purely quantum and stochastic trajectory predictions.
2. EQUATIONS OF MOTION
We consider a two-level atom (with the transition frequency uia and mass ma ) moving in a strong standing laser wave with the modulated frequency ui/[t]. We assume that the depth of modulation is neglible in comparison with the average frequency value {u>/[i]) (but not with the detuning ui/[t] —uJa), and we can therefore
consider the corresponding wave vector k/ a constant. In the absence of spontaneous emission (the atomic excited state must have a long lifetime, or some experimental methods must be used to suppress the deco-herence), the atomic motion can be described by the Hamiltonian
P'2 1
H = + 2h<KU)a ~ UJf^)a'- ~
— hit (<r_ + (T+) cos(k/X), (1)
where a±r_ are the operators of transitions between the atomic excited and ground states (the Pauli matrices), A' and P are the operators of the atomic coordinate and momentum, and Q is the Rabi frequency. This Hamiltonian was used in [33 35], albeit for a constant field without modulation.
We use the following dimensionless normalized quantities: the momentum p = P/hkj, the time r = = i)f, the position x = k/X, the mass m = )nai}/hk'j, and the detuning A[r] = (u;/[r] — u;Q)/ii. We suppose that the field
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